Symmetric Presentations of Sporadic Groups and Related Topics

by John Nicholas Bray

A thesis submitted to the University of Birmingham for the degree of Doctor of Philosophy in the Faculty of Science

School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham, B15 2TT.

30^{th} September, 1997

Acknowledgements

I would like to thank my supervisor Dr. Robert Turner Curtis for his patient and enthusiastic help and for allowing me to follow the avenues of research I found most interesting. I am also grateful to Dr. Christopher W. Parker and Dr. Robert A. Wilson for their helpful discussions. I acknowledge the EPSRC (formerly SERC) for their financial assistance.

I declare that unless stated otherwise, all the work in this thesis submitted for the degree of Ph. D. is my own original work. This may not apply rigorously to results by Curtis, Hammas and Jabbar and to various presentations in Coxeter’s book [11], but I have tried to cite their more important results. This thesis has been written in L^{A}TEX 2ε.

Synopsis

In this thesis, we develop from scratch, the techniques of symmetric presentation and provide presentations for lots of groups, including some of the sporadic simple groups. Most of the results stated in this thesis are proved here, but please note that the proofs of most of the tabulated results of Appendix E do not appear in this thesis, but instead appear in my M. Phil. [3].

Chapter 1 is an introductory chapter which provides motivation for the symmetric presentation project. A few concepts are introduced informally, and we introduce Coxeter groups and monomial groups.

In Chapter 2, we introduce the reader to free products, free amalgamated products and amalgams. This is useful, since progenitors are amalgams. We have included a theorem on the automorphisms of a certain type of free amalgamated product.

In Chapter 3, we introduce the concepts of symmetric presentations in a more formal manner, including such terms as symmetric generator and control subgroup. We also prove some lemmas of a general nature pertaining to symmetric presentations.

In Chapter 4, we present some computer-free approaches to proving the correctness of symmetric presentations. We have also included some examples of manual double coset enumeration in this chapter, and we have given information about how to construct some of the associated geometries starting from the symmetric presentation.

In Chapter 5, we give examples of some of the computational techniques that were utilised in this thesis, for example, to identify the shape of various groups. In addition, the main commands that we used from the computer algebra packages Magma, Cayley, GAP and the MeatAxe are explained.

Chapter 6 describes how to construct matrix representations of groups. It also provides some information about how to find invariant sesquilinear forms.

Chapter 7 details the systematic approach and provides most of the information you would wish to know about these ‘small’ progenitors. The tabulated results associated with this chapter can be found in Appendix E.

Chapter 8 describes a symmetric presentation of the McLaughlin group McL and some other related groups such as U_{4}(3). We find words for all the maximal subgroups in terms of the symmetric generators and elements of the control subgroup N ≅ M_{11}.

In Chapter 9, we describe a symmetric presentation of the Harada–Norton group HN. As by-products of this approach, we obtain explicit 133 × 133 matrices over ℚ(b5) that generate HN, as well as a number of (smaller) representations of smaller subgroups.

In Chapter 10, we describe and (partially) prove a symmetric presentation of the Lyons group Ly. Some of the details of this chapter are absent.

I have included various appendices that deal with such matters as notation, the Hoffman–Singleton graph, a method for finding an involution centraliser, and last, but not least, the tabulated results associated with Chapter 7. Note that we have based our group structure notation and other notation on that given in the Atlas, though we have made a few modifications. Note also, that maps act on the right unless otherwise indicated.

Contents

1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Symmetric generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Example—the symmetric group, S_{n+1} . . . . . . . . . . . . . . . . . 3 1.2.2 Example—the alternating group, A_{n+2} . . . . . . . . . . . . . . . . 3 1.2.3 Example—the Mathieu group, M_{12} . . . . . . . . . . . . . . . . . . 3 1.2.4 Example—the Mathieu group, M_{24} . . . . . . . . . . . . . . . . . . 4 1.2.5 Example—the Chevalley groups . . . . . . . . . . . . . . . . . . . . 4

1.3 Symmetric presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Example—the symmetric group, S_{n+1} . . . . . . . . . . . . . . . . . 5 1.3.2 Example—the alternating group, A_{n+2} . . . . . . . . . . . . . . . . 5 1.3.3 Example—the Mathieu group, M_{12} . . . . . . . . . . . . . . . . . . 6 1.3.4 Example—the Mathieu group, M_{24} . . . . . . . . . . . . . . . . . . 6 1.3.5 Example—the Chevalley groups . . . . . . . . . . . . . . . . . . . . 7

1.4 Monomial transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Coxeter presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Free products and amalgams 12 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Amalgams—definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 The case when we amalgamate two groups . . . . . . . . . . . . . . 14 2.3.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Structure and automorphisms of amalgams . . . . . . . . . . . . . . . . . . 15 2.5 When are two amalgams isomorphic? . . . . . . . . . . . . . . . . . . . . . 17

2.5.1 A useful result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6.1 The case of amalgamating two S_{n}’s over S_{n−1} . . . . . . . . . . . . 19 2.6.2 Amalgamating two M_{12}’s over L_{2}(11) . . . . . . . . . . . . . . . . . 20 2.6.3 Amalgamating two S_{4}’s over D_{8} . . . . . . . . . . . . . . . . . . . . 20 2.6.4 Amalgamating N and H × T over H . . . . . . . . . . . . . . . . . 20 2.6.5 Amalgamating N and (H × T).m over H.m . . . . . . . . . . . . . 21

3 Progenitors 22 3.1 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 A note on monomial groups . . . . . . . . . . . . . . . . . . . . . . 24 3.2.2 Progenitors as amalgams . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 More definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.1 Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.2 N^{ij}, N_{ij} and related notation . . . . . . . . . . . . . . . . . . . . . 25 3.3.3 The coset stabiliser, N^{(w)} . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3.4 Notation for symmetric generators and other objects . . . . . . . . 26 3.3.5 Notation for equivalence relations . . . . . . . . . . . . . . . . . . . 26

3.4 Extra relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5 ⟨x, y, t⟩-presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.6 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6.1 The famous lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.6.2 Results for involutory generators . . . . . . . . . . . . . . . . . . . 28 3.6.3 Collapse of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.7 When are two progenitors isomorphic? . . . . . . . . . . . . . . . . . . . . 31 3.8 Automorphisms of the progenitors . . . . . . . . . . . . . . . . . . . . . . . 31 3.9 How many progenitors? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.9.1 The progenitor 7^{⋆15}:_{m} 3·A_{7} . . . . . . . . . . . . . . . . . . . . . . . 32 3.10 The Campbell and Robertson approach . . . . . . . . . . . . . . . . . . . . 32

4 Some examples 33

4.1 Manual double coset enumeration—examples . . . . . . . . . . . . . . . . . 33 4.1.1 A presentation for PGL_{2}(7) . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2 A presentation for PGL_{2}(11) . . . . . . . . . . . . . . . . . . . . . . 36 4.1.3 A presentation for L_{2}(19) . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1.4 A presentation for A_{7} . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 The symmetric groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 The alternating groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.1 The Moore presentation of A_{n+2} . . . . . . . . . . . . . . . . . . . . 41 4.3.2 Some related presentations . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.3 An amalgam for A_{n+1} . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.4 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Computer packages 47 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Finitely presented groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 Other groups used in this chapter . . . . . . . . . . . . . . . . . . . 48 5.3 Subgroups and order of a group . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Obtaining a permutation representation . . . . . . . . . . . . . . . . . . . . 49 5.5 Cutting a group down to size . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.5.1 Identifying the ‘rough’ shape of our groups . . . . . . . . . . . . . . 51 5.5.2 How to identify automorphisms and/or covers . . . . . . . . . . . . 51 5.5.3 Some general remarks . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.5.4 What happens if a faithful (permutation) representation is hard to

come by? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.6 Some other structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.6.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.6.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.7 The computers used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.8 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 Matrix representations 55 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.2 The basic technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2.1 Choosing H and K . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.2.2 Constructing f_{H}(H) and f_{K}(K) . . . . . . . . . . . . . . . . . . . . 56 6.2.3 Finding f_{H}(L) and f_{K}(L) . . . . . . . . . . . . . . . . . . . . . . . 56 6.2.4 The last two steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2.5 Checking the cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2.6 How do we know we have the answer? . . . . . . . . . . . . . . . . 58

6.3 Our variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.3.1 An example—a 111-dimensional representation of Ly . . . . . . . . 59

Constructing the representations of N ≅ 2·A_{11} . . . . . . . . . . . . 60

6.4 Finding a canonical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4.1 Create seed—CSeed . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4.2 Standard basis—SBasis . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4.3 Finding forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Symplectic and orthogonal forms—the self-dual case . . . . . . . . . 62 Hermitian (or unitary) forms—the not necessarily self-dual case . . 63

6.4.4 An example—a 4-dimensional representation of 2·A_{5} . . . . . . . . 63

7 The systematic approach 65 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 Presentations of our progenitors . . . . . . . . . . . . . . . . . . . . . . . . 65 7.3 Relations we append to our progenitors . . . . . . . . . . . . . . . . . . . . 66

7.3.1 n^{th} order parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.3.2 s-type relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.3.3 Historical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.3.4 The actual cases—involutory generators . . . . . . . . . . . . . . . 71 7.3.5 The actual cases—generators of order 3 (permutation action) …. 72

7.3.6 The actual cases—generators of order 3 (monomial action) . . . . . 72 7.4 A few remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.4.1 Useful subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4.2 General images/presentations . . . . . . . . . . . . . . . . . . . . . 76 7.4.3 The derived series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.4.4 Improper images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.5 Coxeter presentations of our progenitors . . . . . . . . . . . . . . . . . . . 78 7.6 Automorphisms of the progenitors . . . . . . . . . . . . . . . . . . . . . . . 79 7.7 L_{2}(q)’s and PGL_{2}(q)’s in our progenitors . . . . . . . . . . . . . . . . . . . 81

7.7.1 The trivial cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.7.2 The cases 3^{⋆4}:A_{4} and 3^{⋆4}:_{m} S_{4} . . . . . . . . . . . . . . . . . . . . . 82 7.7.3 The cases 2^{⋆4}:A_{4} and 2^{⋆4}:S_{4} . . . . . . . . . . . . . . . . . . . . . . 82 7.7.4 The cases 2^{⋆6}:L_{2}(5) and 3^{⋆6}:_{m} L_{2}(5) . . . . . . . . . . . . . . . . . . 82 7.7.5 The case 3^{⋆3}:S_{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.7.6 The case 3^{⋆2}:_{m} D_{8} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.7.7 The case 2^{⋆3}:A_{3} ≅ 3^{⋆2}:S_{2} ≅ 3^{⋆2}:_{m} S_{2} ≅ 2 ⋆ 3 ≅ L_{2}(ℤ) . . . . . . . . . 83 7.7.8 The case 2^{⋆3}:S_{3} ≅ 3^{⋆2}:_{m} 2^{2} ≅ PGL_{2}(ℤ) . . . . . . . . . . . . . . . . . 84

7.7.8.1 The case L_{2}(q) ≅ PGL_{2}(q) for q even . . . . . . . . . . . . 84 7.7.8.2 The case PGL_{2}(q) for q odd . . . . . . . . . . . . . . . . . 85 7.7.8.3 The case L_{2}(q) for q odd . . . . . . . . . . . . . . . . . . . 85 7.7.8.4 Proof of uniqueness . . . . . . . . . . . . . . . . . . . . . . 87 7.7.8.5 Images U_{3}(q), L_{3}(q) of 2^{⋆3}:S_{3} . . . . . . . . . . . . . . . . 87

7.7.9 The case 3^{⋆3}:_{m} S_{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.7.9.1 Dimension 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.7.9.2 Dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.7.9.3 Some images . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.8 Refinements of previous work . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.8.1 The case G ≅ [2^{9}]:L_{3}(2) . . . . . . . . . . . . . . . . . . . . . . . . 91 7.8.2 The case G ≅ [2^{18}]:3·S_{7} . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.8.3 The case G ≅ [2^{10}3]:S_{5} . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.8.4 The case G ≅ 2.(2^{8}:L_{2}(17)) . . . . . . . . . . . . . . . . . . . . . . 92

7.9 Tabulated results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8 A presentation of the McLaughlin group 93 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.2 The control subgroup and progenitor . . . . . . . . . . . . . . . . . . . . . 93 8.3 Presentations of some subgroups of McL . . . . . . . . . . . . . . . . . . . 94

8.3.1 A presentation of 3_{2}·U_{4}(3) and related presentations . . . . . . . . . 94 8.3.2 A presentation of 3^{4}:M_{10} . . . . . . . . . . . . . . . . . . . . . . . . 96 8.3.3 A presentation of 5{1+2 \atop +}:3 . . . . . . . . . . . . . . . . . . . . . . . . 96

8.4 The McLaughlin group itself . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.4.1 The ‘basic’ presentation . . . . . . . . . . . . . . . . . . . . . . . . 97 8.4.2 When do we have ⟨t_{ij}, π⟩ ≅ 5{1+2 \atop +}:3? . . . . . . . . . . . . . . . . . . 97

8.4.3 Variants of the basic presentation . . . . . . . . . . . . . . . . . . . 97 8.5 A presentation for McL:2 (and 3·McL:2) . . . . . . . . . . . . . . . . . . . 98 8.6 Subgroups of McL generated by subsets of the symmetric generators …. 98

8.6.1 The case ⟨t_{01}, t_{23}⟩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.6.2 The case ⟨t_{01}, t_{12}⟩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.6.3 The case ⟨t_{01}, t_{02}⟩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.6.4 The case ⟨t_{10}, t_{20}⟩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.6.5 The case ⟨t_{01}, t_{10}⟩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.7 Some maximal subgroups of McL and McL:2 . . . . . . . . . . . . . . . . . 100

9 Constructing the Harada–Norton group HN 103 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.2 The Higman–Sims group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.2.1 The double cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.3 The progenitor and extra relation . . . . . . . . . . . . . . . . . . . . . . . 105

9.3.1 The automorphism group . . . . . . . . . . . . . . . . . . . . . . . 106 9.3.2 Some subgroups of HN and HN:2 . . . . . . . . . . . . . . . . . . . 107

9.4 The matrix construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 9.4.1 The 56-dimensional representation of 2·HS:2 . . . . . . . . . . . . . 109 9.4.2 The representation of HN . . . . . . . . . . . . . . . . . . . . . . . 110

9.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

10 The Lyons group 113

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 10.1.1 The graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 10.1.2 Some maximal subgroups . . . . . . . . . . . . . . . . . . . . . . . 115 10.1.3 The presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

A Notation 119 A.1 Some conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.2 Notation for particular groups . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.3 Notation for particular group presentations . . . . . . . . . . . . . . . . . . 121 A.4 Notation for particular sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.5 General group theoretic notation . . . . . . . . . . . . . . . . . . . . . . . 122 A.6 Notation pertaining to progenitors . . . . . . . . . . . . . . . . . . . . . . . 124 A.7 Atomic algebraic irrationalities (over ℂ) . . . . . . . . . . . . . . . . . . . 125 A.8 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

B Some Coxeter groups 126

B.1 Proofs of some of the basic presentations . . . . . . . . . . . . . . . . . . . 126 B.1.1 The cases (2, 2, n) ≅ D_{2n} and (2, 3, 2) ≅ D_{6} ≅ S_{3} . . . . . . . . . . . 126 B.1.2 The case (2, 3, 3) ≅ A_{4} ≅ L_{2}(3) . . . . . . . . . . . . . . . . . . . . 126 B.1.3 The case (2, 3, 4) ≅ S_{4} ≅ PGL_{2}(3) . . . . . . . . . . . . . . . . . . . 126 B.1.4 The case (2, 3, 5) ≅ A_{5} . . . . . . . . . . . . . . . . . . . . . . . . . 127 B.1.5 The case (2, 4, 5; 3) ≅ S_{5} . . . . . . . . . . . . . . . . . . . . . . . . 127 B.1.6 The case (2, 3, 7; 4) ≅ L_{3}(2) . . . . . . . . . . . . . . . . . . . . . . 127 B.1.7 The case (3, 3 | 4, 5) ≅ 3·A_{6} . . . . . . . . . . . . . . . . . . . . . . 128

B.2 The finite Coxeter diagram groups . . . . . . . . . . . . . . . . . . . . . . . 128 B.3 The groups G^{p,q,r} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 B.4 The groups (p, q, r) and (p, q, r; s) . . . . . . . . . . . . . . . . . . . . . . . 129

C The Hoffman–Singleton graph 131 C.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 C.2 Some objects associated with S_{6} . . . . . . . . . . . . . . . . . . . . . . . . 131 C.3 Proof of existence (construction) . . . . . . . . . . . . . . . . . . . . . . . . 132 C.4 The automorphism group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

D A computational trick 134

D.1 Finding the centraliser of an involution . . . . . . . . . . . . . . . . . . . . 134 D.1.1 The first method (the dihedral group method) . . . . . . . . . . . . 134 D.1.2 The second method . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

D.1.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 D.1.2.2 Some comments . . . . . . . . . . . . . . . . . . . . . . . . 135 D.1.2.3 An example—U_{6}(2) . . . . . . . . . . . . . . . . . . . . . 135

E The tables 137 E.1 Tabulated results—notation . . . . . . . . . . . . . . . . . . . . . . . . . . 137 E.2 Accuracy of the tabulated results . . . . . . . . . . . . . . . . . . . . . . . 138 E.3 Some sample proofs of the tabulated results . . . . . . . . . . . . . . . . . 138

E.3.1 The images S_{7} and 3·S_{7} of 2^{⋆4}:S_{4} . . . . . . . . . . . . . . . . . . . 138 E.3.2 The image ^{2}F_{4}(2)′ × 2 of 2^{⋆4}:S_{4} . . . . . . . . . . . . . . . . . . . . 139 E.3.3 The image 2·(L_{2}(19)^{2}):2^{2} of 2^{⋆4}:S_{4} . . . . . . . . . . . . . . . . . . 139 E.3.4 The image 2^{1+6}:U_{3}(3) of 3^{⋆4}:S_{4} . . . . . . . . . . . . . . . . . . . . 139 E.3.5 The image 2^{8_{2}} :A_{9} of 3^{⋆4}:S_{4} . . . . . . . . . . . . . . . . . . . . . . . 140 E.3.6 The image L_{4}(3):2_{2} of 3^{⋆5}:S_{5} . . . . . . . . . . . . . . . . . . . . . . 140

F Some matrix representations 162

Bibliography 168

Chapter 1

Introduction

1.1 Motivation

Groups arise naturally in many areas of mathematics. For example, the rotational symmetries of an icosahedron form a group (in this case the alternating group A_{5}). The set of automorphisms of a graph also forms a group. (An automorphism of a graph is a permutation of the vertices that preserves adjacency.) This is of particular interest if the group is vertex transitive, that is, the graph ‘looks the same’ from the point of view of any vertex.

In this thesis, we shall be primarily concerned with finite groups. In particular, we shall be concerned with finite simple groups since all finite groups can be ‘built up’ from these. With the Classification of Finite Simple Groups, henceforth abbreviated as CFSG, a finite simple group belongs to one of the following categories:

• The cyclic groups C_{p} for p prime,

• The alternating groups A_{n} for n ⩾ 5,

• The classical groups: L_{n}(q) (n ⩾ 2), U_{n}(q) (n ⩾ 3), S_{2m}(q) (m ⩾ 2), O_{2m+1}(q) (m ⩾ 3), O_{2m}^{+}(q) (m ⩾ 4) and O_{2m}^{−}(q) (m ⩾ 4), where q ⩾ 2 is a prime power,

• Ten classes of exceptional groups of Lie type,

• 26 sporadic simple groups that do not belong to the above classes, and

• The Tits group ^{2}F_{4}(2)′.

Notice that finitely many groups in the above list are not simple. The Tits group, ^{2}F_{4}(2)′, is usually counted as an exceptional group of Lie type rather than as being sporadic.

Of the simple groups, the groups C_{p} and A_{n} are trivial to construct. The classical groups appear as automorphisms of a vector space, sometimes preserving a sesquilinear form. The groups of Lie type (including the classical groups) have standard constructions as matrix groups, see Carter [8] for example. Some of the exceptional groups of Lie type, for example G_{2}(q), admit constructions that closely resemble constructions of the classical groups. This leaves the sporadic groups which have been constructed by a variety of ad hoc methods.

The Mathieu groups were constructed as highly transitive permutation groups in 1861 (M_{11} and M_{12}) and 1873 (M_{22}, M_{23} and M_{24}). The remaining sporadic simple groups were discovered more recently—in the 1960s and 1970s. Some of these arise as subgroups of index at most 2 in the automorphism groups of rank 3 graphs, for example HS, J_{2}, McL, Suz, Fi_{22}, Fi_{23} and Fi′_{24}. Others arise from the Leech Lattice; for example the Conway groups Co_{1}, Co_{2} and Co_{3}. However, some of the groups, for example J_{1}, J_{2}, J_{3}, J_{4}, He and Ly, were initially defined as (simple) groups having certain involution centralisers. Moreover, some of these still lack any kind of ‘satisfactory’ definition and/or construction.

One of the aims of symmetric generation (see below) is to provide a uniform way of constructing the sporadic groups. We hope that this will provide new insights into the structure of those groups for which many pleasing constructions already exist. But if the techniques we develop are to be of much use, the infinite families of simple groups (in particular the classical groups) should be open to the techniques of symmetric generation. Indeed, we find this to be the case.

Note that earlier work in this area has been done by R. T. Curtis, A. M. A. Hammas and A. Jabbar, the latter two both having been research students of R. T. Curtis. In this chapter, we aim to illustrate the relationship between symmetric presentations and other presentations, for example Coxeter presentations. We shall see that we may express various classical presentations of H. S. M. Coxeter and E. H. Moore in symmetric presentation format. The symmetric presentations of C. M. Campbell and E. F. Robertson are viewed from this perspective. Thus they are not symmetric presentations in our parlance, but they are very close relatives.

We start by introducing the concepts of symmetric generation, symmetric presentation, a progenitor and monomial groups. We define some of these concepts in a more formal manner in Chapter 3.

1.2 Symmetric generation

Let G be a group generated by (proper) subgroups:

T_{0}, T_{1}, …, T_{n−1},

each isomorphic to T such that

N := N_{G}({T_{0}, T_{1}, …, T_{n−1}}),

acts transitively on the T_{i}. Then we say that G is symmetrically generated by the T_{i}. We call N the control subgroup of G. Thus G is a homomorphic image of the progenitor:

T^{⋆n}:N, where T^{⋆n} denotes the free product of n copies of T and A:B denotes the semidirect product of (groups) A and B. If the T_{i} are cyclic, so T_{i} = ⟨t_{i}⟩ say, we define the symmetric generators to be the images of t_{0} under the conjugation action of N. (It is convenient and useful to define symmetric generators in other circumstances as well.) We give some examples below.

1.2.1 Example—the symmetric group, S_{n+1}

It is well-known that the symmetric group G ≅ S_{n+1} (acting on {0, …, n}) is generated by the transpositions:

T = {t_{0} = (0, n), t_{1} = (1, n), …, t_{n−1} = (n−1, n)}.

Writing T_{i} for ⟨t_{i}⟩, we see that the control subgroup N is S_{n} acting on {0, …, n−1}. Thus S_{n+1} is a homomorphic image of the progenitor

2^{⋆n}:S_{n}.

Note that in this case the t_{i} = (i, n) are the symmetric generators.

1.2.2 Example—the alternating group, A_{n+2}

It is also well-known that the alternating group G ≅ A_{n+2} is generated by the 3-cycles:

T = {t_{0} = (0, n, n+1), t_{1} = (1, n, n+1), …, t_{n−1} = (n−1, n, n+1)}.

Here the control subgroup N := N_{G}(T̄) is S_{n} with orbits {0, …, n−1} and {n, n+1}, where T̄ := {T_{0}, T_{1}, …, T_{n−1}} with T_{i} = ⟨t_{i}⟩ as before. The action of N on the t_{i} is more complicated than before, since the outer elements of N permute and invert the t_{i}. In this case, we say that N acts monomially on the t_{i} (see Section 1.4). The symmetric generators here are the t_{i} and the t_{i}^{−1}. We have that A_{n+2} is a homomorphic image of the progenitor

3^{⋆n}:_{m} S_{n},

where :_{m} indicates that the action of N on the T_{i} is strictly monomial (that is, N acts monomially, but not as permutations, on the T_{i}).

1.2.3 Example—the Mathieu group, M_{12}

The next two examples are taken from a paper of Curtis [13]. It was the discovery of these ways of generating M_{12} and M_{24} and the symmetric presentations (see Section 1.3) associated with them which motivated Curtis’s interest in this area. We have that M_{12} is generated by five cyclic subgroups of order 3 permuted by a control subgroup N ≅ A_{5}. We may take:

\begin{matrix}x ∼ & (1, 3, 9, 5, 4)(2, 6, 7, X, 8), \\ y ∼ & (∞, 6, 7)(0, 3, 9)(1, 8, 5)(2, 4, X), \\ and t ∼ & (∞, 8, X)(0, 3, 9)(1, 4, 7)(2, 6, 5), \end{matrix}

where N = ⟨x, y⟩ ≅ A_{5} and t ∼ t_{0} is a symmetric generator. (Here x ∼ (0, 1, 2, 3, 4) and y ∼ (4, 2, 1) in their action on the symmetric generators.) Curtis showed that it is possible to label the faces of a dodecahedron in such a way that the control subgroup N is just the rotation group of the dodecahedron, and t effects an anticlockwise rotation of the faces adjoining each of 4 tetrahedrally placed vertices. See Curtis [14] for more details.

1.2.4 Example—the Mathieu group, M_{24}

The Mathieu group M_{24} is a homomorphic image of a progenitor 2^{⋆7}:L_{3}(2), where the control subgroup N ≅ L_{3}(2) is the maximal octern subgroup of M_{24}. We may take N = ⟨x, y⟩ where x ∼ (0, 1, 2, 3, 4, 5, 6) and y ∼ (2, 6)(4, 5) in their actions on the symmetric generators. The realisations of x, y and t as permutations of 24 points are given below:

\begin{matrix}x ∼ & (0, 8, 18, 14, 3, 15, 20)(1, 9, 6, 19, 12, 21, 5)(2, 11, 16, 10, 7, 4, 13), \\ y ∼ & (∞, 7)(0, 19)(1, 20)(2, 15)(3, 18)(4, 10)(5, 12)(6, 16)(8, 11)(9, 13)(14, 17)(21, 22), \\ and t ∼ & (∞, 20)(0, 5)(1, 7)(2, 15)(3, 18)(4, 16)(6, 10)(8, 11)(9, 14)(12, 19)(13, 17)(21, 22). \end{matrix}

There is a ‘polyhedral’ interpretation of this situation analogous to that for the M_{12} case. However, the ‘polyhedron’ is now a genus 3 object with 24 heptagonal faces meeting 3 to a vertex. Again, see Curtis [14] for more details.

1.2.5 Example—the Chevalley groups

Let G be a (untwisted) Chevalley group. Then G is symmetrically generated by the root subgroups, whose control subgroup is the monomial subgroup in the sense of Carter [8, page 101]. Fortuitously, the control subgroup is the N of the (‘standard’) (B, N)-pair generating such a group, see Carter [8, page 107]. Thus, two subgroups whose standard notation is N happen to coincide in this instance. From now on we shall restrict our attention to the case when G is (a decoration of) A_{n−1}(q) (for n ⩾ 2). Specifically, we shall consider G ≅ GL_{n}(q) (for n ⩾ 2).

So let F = 𝔽_{q} and let V = F^{n} with basis {e_{1}, e_{2}, …, e_{n}}. Then for λ ∈ F and 1 ⩽ i, j ⩽ n with i ≠ j, we define the element t_{ij}(λ) ∈ GL(V) by:

t_{ij}(λ) = I + λE_{ij},

where I is the identity and the E_{ij} are the elementary matrices with a 1 in the (i, j)^{th} place and zeros elsewhere. Furthermore, we define T_{ij} = {t_{ij}(λ) | λ ∈ F } and t_{ij} = t_{ij}(1) for all i ≠ j. Thus the T_{ij} are transvection subgroups.

Here, we take the group of monomial matrices as the control subgroup N and the T_{ij} as the T ’s on which N acts. Thus, since T_{ij} ≅ (𝔽_{q}, +) ≅ E_{q} and N ≅ (𝔽_{q}^{×}, ·) ≀ S_{n} ≅ C_{q−1} ≀ S_{n}, it follows that G ≅ GL_{n}(q) is a homomorphic image of the progenitor

E_{q}^{⋆n(n−1)}:_{m} (C_{q−1} ≀ S_{n}).

Observe, however, that the scalars of N act trivially on the T_{ij}. Thus, factoring out these scalars, we see that PGL_{n}(q) is a homomorphic image of a progenitor of shape

E_{q}^{⋆n(n−1)}:_{m} ((C_{q−1})^{n−1}:S_{n}).

Note that there are analogues of this for the other Chevalley groups.

1.3 Symmetric presentations

Given that G is a homomorphic image of a progenitor T^{⋆n}:N, let ϕ be a homomorphism of T^{⋆n}:N onto G. (Note that ϕ need not be unique, or even ‘essentially’ unique.) Then if K = ker ϕ, we have that (T^{⋆n}:N)/K ≅ G. While it is quite difficult to exhibit all elements of K explicitly, it is quite reasonable (for small G) to find elements g_{1}, g_{2}, …, g_{m} ∈ T^{⋆n}:N such that the normal closure (in T^{⋆n}:N) of {g_{1}, g_{2}, …, g_{m}} is K. Thus we may write:

G ≅ \frac{T^{⋆n}:N}{g_{1} = g_{2} = ⋯ = g_{m} = 1},

and refer to this as a symmetric presentation for G. Note that since T^{⋆n}:N is a semidirect product, the elements g_{i} have unique representations of the form:

g_{i} = π_{i}w_{i},

where π_{i} ∈ N and w_{i} is an element of T^{⋆n}. (Note also that g_{i}g_{j} = π_{i}π_{j}.w_{i}^{π_{j}}w_{j}.) Thus we may write the above symmetric presentation in the form:

\frac{T^{⋆n}:N}{π_{1}w_{1} = π_{2}w_{2} = ⋯ = π_{m}w_{m} = 1} ≅ G.

We now list symmetric presentations corresponding to the examples of symmetric generation discussed previously. We prove some of them later.

1.3.1 Example—the symmetric group, S_{n+1}

We may obtain S_{n+1} from 2^{⋆n}:S_{n} by factoring out by [(0, 1)t_{0}]^{3} = 1. We write this as follows:

\frac{2^{⋆n}:S_{n}}{[(0, 1)t_{0}]^{3} = 1} ≅ S_{n+1}.

Note that the relation we factor out by is equivalent to t_{0}t_{1}t_{0} = (0, 1).

1.3.2 Example—the alternating group, A_{n+2}

The relevant symmetric presentations are:

\frac{3^{⋆n}:_{m} S_{n}}{t_{0}t_{1}^{−1}t_{0} = (0, 1)} ≅ A_{n+2},

and

\frac{3^{⋆n}:_{m} S_{n}}{t_{0}t_{1}t_{0}t_{1} = 1} ≅ A_{n+2} × S_{n}.

In the first symmetric presentation, the element denoted (0, 1) has action (t_{0}, t_{1}^{−1})(t_{i}, t_{i}^{−1}) (for i ⩾ 2) on the symmetric generators, and the extra relation appended to our progenitor is equivalent to [(0, 1)t_{0}]^{3} = 1. The second symmetric presentation restricted to ⟨T ⟩ is essentially equivalent to the following presentation:

⟨t_{0}, …, t_{n−1} | t_{i}^{3} = 1 = (t_{i}t_{j})^{2} ∀ i≠j⟩.

This was shown be a presentation for A_{n+2} by E. H. Moore about a century ago and is one of the earliest examples of a symmetrically presented group (but not quite a symmetric presentation in our sense). Symmetric presentations in the sense of Campbell and Robertson are of this latter type.

1.3.3 Example—the Mathieu group, M_{12}

In this case, the shortest (non-trivial) word in the symmetric generators that lies in N is [t_{i}t_{j}^{−1}]^{2} for i ≠ j. We have that

\frac{3^{⋆5}:A_{5}}{(4, 2, 1) = [t_{0}t_{3}^{−1}]^{2}} ≅ M_{12} × 3,

where the central 3-group is generated by [(0, 1, 2, 3, 4)t_{0}]^{8}. We may extend the control subgroup to S_{5} where the outer elements of S_{5} permute and invert the symmetric generators. In this case we have:

\frac{3^{⋆5}:_{m} S_{5}}{(4, 2, 1) = [t_{0}t_{3}^{−1}]^{2}} ≅ (M_{12} × 3):2,

where (4, 2, 1) = [t_{0}t_{3}^{−1}]^{2} is equivalent to [(0, 1, 2)(3, 4)t_{3}]^{4} = 1 (but is not equivalent to [(0, 1, 2)(3, 4)t_{3}^{−1}]^{4} = 1). As before, the normal 3-group is generated by [(0, 1, 2, 3, 4)t_{0}]^{8}. A single relation such as (4, 2, 1) = [t_{0}t_{3}^{−1}]^{2}[(0, 1, 2, 3, 4)t_{0}^{−1}]^{8} suffices to define M_{12} or M_{12}:2 depending on the case we are considering.

1.3.4 Example—the Mathieu group, M_{24}

A symmetric presentation of M_{24} as a homomorphic image of 2^{⋆7}:L_{3}(2) is given by

\frac{2^{⋆7}:L_{3}(2)}{[(2, 6, 4, 5)(0, 3)t_{2}]^{11} = [(2, 6)(4, 5)t_{0}t_{2}]^{4} = 1, (2, 6)(4, 5) = [t_{1}t_{3}]^{3}} ≅ M_{24}.

We note that the relations force ⟨t_{2}, t_{4}, t_{5}, t_{6}⟩ ≅ L_{2}(23). (This is not obvious.) The presentations given here for both M_{12} and M_{24} are taken from Curtis [13].

1.3.5 Example—the Chevalley groups

Let the notation for groups, vector spaces, etc. be as in Section 1.2.5. Then we have:

\frac{E_{q}^{⋆n(n−1)}:_{m} (C_{q−1} ≀ S_{n})}{t_{12}t_{21}^{−1}t_{12} = π, [t_{12}, t_{13}] = [t_{21}, t_{31}] = 1 = t_{13}^{−1}[t_{12}, t_{23}], [t_{12}, t_{34}] = 1} ≅ GL_{n}(q),

where π = (1, 2, ¯1, ¯2) acts on V be taking (e_{1}, e_{2}, e_{i} (i ⩾ 3)) to (e_{2}, −e_{1}, e_{i} (i ⩾ 3)) and relations involving subscripts (strictly) greater than n are omitted. Note that the (omitted) relation [t_{21}, t_{32}] = t_{31}^{−1} is a consequence of [t_{12}, t_{23}] = t_{13} and so is redundant. Similarly, we have:

\frac{E_{q}^{⋆n(n−1)}:_{m} ((C_{q−1})^{n−1}:S_{n})}{t_{12}t_{21}^{−1}t_{12} = π, [t_{12}, t_{13}] = [t_{21}, t_{31}] = 1 = t_{13}^{−1}[t_{12}, t_{23}], [t_{12}, t_{34}] = 1} ≅ PGL_{n}(q).

These are closely related to the Steinberg presentations of Chevalley groups, see Carter [8, page 190]. Thus analogues of these results exist for all the (untwisted) Chevalley groups. In the future, we intend to investigate these configurations further, placing them in a classical context and extending the results to cover the twisted groups of Lie type. We also intend to investigate other generic configurations/presentations which are not (clearly) related to the above.

1.4 Monomial transformations

Let R be a ring with 1.^{1} Let M be an n × n matrix with entries in R (with respect to a spanning set {e_{0}, …, e_{n−1}} of R^{n}). Then we say that M is a monomial matrix if M has precisely one non-zero entry in each row and column and the non-zero entries are units of R. If M is a monomial matrix, then M has a unique decomposition as:

M = M_{d}.M_{p},

where M_{d} is a diagonal matrix and M_{p} is a permutation matrix (which we shall often regard as simply being a permutation).

We define a monomial group to be a group of monomial matrices (under multiplication). Let N be a monomial group. Then N_{p}, the corresponding permutation group of N, is defined by:

N_{p} = {M_{p} | M ∈ N}.

Suppose that N_{p} is transitive. Then we may obtain the representation of N on R^{n} by inducing up the ‘1-dimensional’ representation of N_{0} = Stab_{N}(⟨e_{0}⟩) on ⟨e_{0}⟩ to N. If this ‘1-dimensional’ representation is trivial (that is, if N_{0} centralises ⟨e_{0}⟩), then we just obtain a permutation representation. Some examples are given below in Section 1.4.2.

1Usually, R will be a commutative ring with 1 or even a field, but there is nothing to stop R being, for example, the group ring ℤG (or even ℤ_{2}G) for some arbitrary group G.

1.4.1 Notes

Let Mo_{n}(R) be the full monomial group of n×n matrices over R. Then two representations ρ, ρ′ : G → Mo_{n}(R) are said to be monomially equivalent if there exists h ∈ Mo_{n}(R) such that (gρ)^{h} = gρ′ for all g ∈ G.

We do not restrict ourselves to the case when R is commutative. Let H ⩽ G be groups and let T be a right transversal of H in G. Define j = j(i, g) to be the unique j such that Ht_{i}g = Ht_{j}. Let ρ : H → R be a representation. Then we define ind_{H}^{G} ρ, the induced representation of ρ to G, by requiring that the matrix of ind_{H}^{G} ρ(g) is monomial of degree |G : H| with (t_{i}gt_{j}^{−1})ρ in the (i, j)^{th} place and zeros elsewhere. The possibilities for ind_{H}^{G} ρ are monomially equivalent.

A more general definition of induced representation in the case when R is noncommutative is under investigation.

1.4.2 Examples

Example 1.4.1. Let N ≅ 3·A_{6}, the (proper) triple cover of A_{6}. Then N can be shown to have presentation:

⟨x, y | x^{3} = y^{3} = (xy)^{4} = (xy^{−1})^{5} = 1⟩ ≅ (3, 3 | 4, 5),

where x ∼ (1, 3, 4) and y ∼ (0, 1, 2)(3, 4, 5) in the image A_{6}. We find that N_{0} = ⟨x, x^{y}⟩ ≅ A_{5} × 3 has index 6 in N. Thus N_{0} has 3 linear representations (over a field that has primitive cube roots of unity). Inducing up the trivial representation (of N_{0} to N) gives rise to a permutation representation of N on 6 points which is really a representation of the image A_{6} and so is not faithful for N ≅ 3·A_{6}. Inducing up a non-trivial linear representation of N_{0} to N gives rise to a faithful irreducible representation of N (except in characteristic 2 when it is not irreducible) with matrices:

x ∼ [ \begin{matrix}ω & · & · & · & · & · \\ · & · & · & 1 & · & · \\ · & · & ω̄ & · & · & · \\ · & · & · & · & 1 & · \\ · & 1 & · & · & · & · \\ · & · & · & · & · & 1 \end{matrix} ] and y ∼ [ \begin{matrix}· & 1 & · & · & · & · \\ · & · & 1 & · & · & · \\ 1 & · & · & · & · & · \\ · & · & · & · & 1 & · \\ · & · & · & · & · & 1 \\ · & · & · & 1 & · & · \end{matrix} ] ,

when the induction is carried out with respect to the transversal {1, y, y^{2}, yx, yxy, yxy^{2}}. Here ω^{3} = 1 ≠ ω and ω̄ = ω^{2} = −1 − ω. We have that [x, y]^{−5} = ωI_{6}, so that in any characteristic but 3, the central elements of order 3 act non-trivially. (In characteristic 3, this latter representation is the same as the permutation representation of N and is not faithful.) Another monomial representation may be obtained by interchanging ω and ω̄ while fixing 1 and 0. In this particular case, this new representation is essentially equivalent to the old one, that is, the new representation, which is generated by matrices (x̄, ȳ = y), is monomially equivalent to the representation generated by the matrices (xα, yα) where α is an outer automorphism of N. Note that we may sometimes abbreviate the above generators to x ∼ (0, 0^{ω}, 0ω̄)(1, 3, 4)(2, 2ω̄, 2^{ω}) or (0, ¯0, ¯ ¯0)(1, 3, 4)(2, ¯ ¯2, ¯2) and y ∼ (0, 1, 2)(3, 4, 5).

Example 1.4.2. Now let N ≅ 3·S_{6}. A subgroup H = N_{0} of index 6 in N has shape (A_{5} × 3):2 ≅ \frac{1}{2}(S_{5} × S_{3}). In this case, we require our ‘linear’ representation of H to have kernel N^{0} ≅ A_{5}. But then N_{0}/N^{0} ≅ S_{3} and so the ring over which we take the representation must be non-commutative. Here, we shall let R be the group ring ℤS_{3}. A presentation for 3·S_{6} is given by:

⟨x, y | x^{6} = y^{2} = 1 = [x, y]^{3}, (xy)^{5} = [x^{2}, y]^{2}⟩.

where x ∼ (0, 1, 2, 3, 4, 5) and y ∼ (4, 5) in the image S_{6}. We find that N_{0} = ⟨y, x^{−1}yx^{2}⟩ ≅ (A_{5} ×3):2. In R, we let S_{3} = ⟨a, b | a^{2} = b^{3} = (ab)^{2} = 1⟩. Then we find that the monomial transformations shown below satisfy the above presentation:

x ∼ diag(a, a, a, a, a, a) × (0, 1, 2, 3, 4, 5)
and y ∼ diag(a, ab^{2}, ab^{2}, ab, ab^{2}, ab^{2}) × (4, 5).

Written in terms of matrices (and noting that ab^{2} = ba), this becomes:

x ∼ [ \begin{matrix}· & a & · & · & · & · \\ · & · & a & · & · & · \\ · & · & · & a & · & · \\ · & · & · & · & a & · \\ · & · & · & · & · & a \\ a & · & · & · & · & · \end{matrix} ] and y ∼ [ \begin{matrix}a & · & · & · & · & · \\ · & ba & · & · & · & · \\ · & · & ba & · & · & · \\ · & · & · & ab & · & · \\ · & · & · & · & · & ba \\ · & · & · & · & ba & · \end{matrix} ] .

If we let u = −1, over any field we have:

a ∼ [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] and b ∼ [ \begin{matrix}0 & u \\ 1 & u \end{matrix} ] ,

and so we can write:

x ∼ [ \begin{matrix}· & · & 0 & 1 & · & · & · & · & · & · & · & · \\ · & · & 1 & 0 & · & · & · & · & · & · & · & · \\ · & · & · & · & 0 & 1 & · & · & · & · & · & · \\ · & · & · & · & 1 & 0 & · & · & · & · & · & · \\ · & · & · & · & · & · & 0 & 1 & · & · & · & · \\ · & · & · & · & · & · & 1 & 0 & · & · & · & · \\ · & · & · & · & · & · & · & · & 0 & 1 & · & · \\ · & · & · & · & · & · & · & · & 1 & 0 & · & · \\ · & · & · & · & · & · & · & · & · & · & 0 & 1 \\ · & · & · & · & · & · & · & · & · & · & 1 & 0 \\ 0 & 1 & · & · & · & · & · & · & · & · & · & · \\ 1 & 0 & · & · & · & · & · & · & · & · & · & · \end{matrix} ] and y ∼ [ \begin{matrix}0 & 1 & · & · & · & · & · & · & · & · & · & · \\ 1 & 0 & · & · & · & · & · & · & · & · & · & · \\ · & · & u & 0 & · & · & · & · & · & · & · & · \\ · & · & u & 1 & · & · & · & · & · & · & · & · \\ · & · & · & · & u & 0 & · & · & · & · & · & · \\ · & · & · & · & u & 1 & · & · & · & · & · & · \\ · & · & · & · & · & · & 1 & u & · & · & · & · \\ · & · & · & · & · & · & 0 & u & · & · & · & · \\ · & · & · & · & · & · & · & · & · & · & u & 0 \\ · & · & · & · & · & · & · & · & · & · & u & 1 \\ · & · & · & · & · & · & · & · & u & 0 & · & · \\ · & · & · & · & · & · & · & · & u & 1 & · & · \end{matrix} ] .

Example 1.4.3. Here we let N ≅ L_{3}(2). Then a subgroup H = N_{0} of index 7 in N has shape S_{4} and homomorphic images 1, 2, S_{3} and S_{4}. Here, we require our linear representation to have kernel N^{0} ≅ 2^{2} ≅ V_{4}. Then, as in the previous example, we have N^{0}/N_{0} ≅ S_{3} and so we choose R = ℤS_{3}. Then N has presentation:

⟨x, y | x^{7} = y^{2} = (yx)^{3} = [x, y]^{4} = 1⟩ ≅ (2, 3, 7; 4),

where x ∼ (0, 1, 2, 3, 4, 5, 6), y ∼ (2, 6)(4, 5) and N_{0} = ⟨y, y^{x^{−1}}, y^{x^{−3}}⟩. If we let the S_{3} of R be generated by a and b subject to a^{2} = b^{3} = (ab)^{2} = 1 then the representation of N_{0} we wish to induce up takes values 1, a and ab on y, y^{x^{−1}} and y^{x^{−3}} respectively. Thus, inducing up with respect to the transversal {1, x, x^{2}, x^{3}, x^{4}, x^{5}, x^{6}} we obtain the following monomial matrices:

x ∼ [ \begin{matrix}· & 1 & · & · & · & · & · \\ · & · & 1 & · & · & · & · \\ · & · & · & 1 & · & · & · \\ · & · & · & · & 1 & · & · \\ · & · & · & · & · & 1 & · \\ · & · & · & · & · & · & 1 \\ 1 & · & · & · & · & · & · \end{matrix} ] and y ∼ [ \begin{matrix}1 & · & · & · & · & · & · \\ · & a & · & · & · & · & · \\ · & · & · & · & · & · & a \\ · & · & · & ab & · & · & · \\ · & · & · & · & · & b̄ & · \\ · & · & · & · & b & · & · \\ · & · & a & · & · & · & · \end{matrix} ] ,

where b̄ = b^{−1}. Let F be a field in which 1 ≠ −1 and let ϕ : R → F be a homomorphism defined by aϕ = −1, bϕ = 1 and mϕ = m for all m ∈ ℤ. Then if we apply ϕ to the entries of the above matrices we obtain:

x̄ ∼ diag(1, 1, 1, 1, 1, 1, 1) × (0, 1, 2, 3, 4, 5, 6) = (0, 1, 2, 3, 4, 5, 6)
and ȳ ∼ diag(1, −1, −1, −1, 1, 1, −1) × (2, 6)(4, 5) = (1, ¯1)(3, ¯3)(2, ¯6)(4, 5),

which defines a 7-dimensional faithful irreducible monomial representation of N ≅ L_{3}(2) over F (not of characteristic 2).

1.5 Coxeter presentations

A Coxeter group is a group generated by a set of involutions (that is elements of order 2) {x_{i}} subject to relations (x_{i}x_{j})^{n}. Usually such groups are infinite and we add extra relations to obtain finite groups. (These images are also referred to as Coxeter groups.) We represent (finitely generated) Coxeter groups diagrammatically by taking a set of nodes (one for each generator) labelled with the generators {x_{i}} such that:

if o(x_{i}x_{j}) = 2 then x_{i} and x_{j} are unjoined,
if o(x_{i}x_{j}) = 3 then x_{i} and x_{j} are joined by an unmarked edge, and
if o(x_{i}x_{j}) = n then x_{i} and x_{j} are joined by an edge marked n,

where in the last case n ⩾ 4, n = ∞ or n is indeterminate.

Let G have Coxeter presentation as shown below:

[figure: N^{0}; g; t_{0}; m_{1}; m_{2}]

where the leftmost circle contains an arbitrary number of involutory generators that generate N^{0} and commute with t_{0}. Let N = ⟨N^{0}, g⟩ and let n = |N : N^{0}|. Then G is a homomorphic image of the progenitor

2^{⋆n}:N.

Note that in the above diagram, we allow more than one edge to connect g into the generators of N^{0}.

As an example, consider the following presentation that is given on page 11 of the Atlas [10] and which we believe is due to L. H. Soicher:

[figure: ⟨  | (abc)^{5} = (bcd)^{5} = 1⟩ ≅ L_{2}(19).; a; b; c; d; 5; 5]

Then the relations between a, b and c force ⟨a, b, c⟩ ≅ A_{5} ≅ L_{2}(5) (or a homomorphic image thereof) where a ∼ (1, 4)(2, 3), b ∼ (0, 3)(1, 2) and c ∼ (∞, 0)(1, 4). Thus if we let N = ⟨a, b, c⟩ ≅ L_{2}(5), N^{0} = ⟨a, b⟩ ≅ D_{10}, g = c, t_{∞} = d and G = ⟨a, b, c, d⟩, we find that G ≅ L_{2}(19) is a homomorphic image of the progenitor 2^{⋆6}:L_{2}(5). Now the only relations of G that do not define the progenitor 2^{⋆6}:L_{2}(5) are (bcd)^{5} = (cd)^{5} = 1 and so we have:

\frac{2^{⋆6}:L_{2}(5)}{[(∞, 0, 3)(1, 2, 4)t_{∞}]^{5} = [(∞, 0)(1, 4)t_{∞}]^{5} = 1} ≅ L_{2}(19),

where the relation [(∞, 0)(1, 4)t_{∞}]^{5} = 1, which corresponds to (cd)^{5} = 1, is redundant.

A more sophisticated approach, which we will discuss briefly later, produces a relationship between Coxeter groups and (monomial) progenitors with non-involutory symmetric generators.

Free products and amalgams

2.1 Motivation

Throughout this thesis, we will be concerned with progenitors (which we will define in Chapter 3) and the homomorphic images thereof. One way of viewing a progenitor T^{⋆n}:N is as a split extension of a free product of n copies of T by a (control) group N permuting them around. Free products are covered adequately in many (general) group theory texts, for example Chapter 7 of Hall [21]. Therefore we will only say enough about free products (and presentations) so as to be able to state our conventions pertaining to such concepts as length. We do this in Section 2.2.

Another way of viewing a progenitor is as a free amalgamated product A ⋆_{C} B. These entities are less well covered in Hall [21] and similar texts. Thus, the aim of this chapter is to introduce the concepts of an amalgam and a free amalgamated product and related concepts, so that we may state, and sometimes prove, sufficient results in order to obtain a (rudimentary) classification of the automorphisms and isomorphism classes of a free amalgamated product A ⋆_{C} B (and its related amalgam) in the case when A and B are finite. We will use these results in Chapter 3 to calculate the outer automorphism group of a given progenitor, and also to establish necessary and sufficient (and easily checked) conditions for two progenitors to be isomorphic.

Finally, in Section 2.6, we give some examples of amalgams, and include for example, the number of isomorphism classes of such an amalgam.

Note that although amalgams and free products are (usually) infinite, we shall only require the concepts when the groups we are amalgamating (or taking the free products of) are finite. However, some (but by no means all), of the concepts and/or results of this chapter apply to arbitrary groups.

2.2 Notation and conventions

Essentially, we expect the reader to be familiar with the concepts of this section. However, we have given some rather loose ‘definitions’ of various concepts so that we may introduce some notation.

Definition 2.2.1. Let G be a group and let S ⊆ G. We define the normal closure of S in G as follows:

⋂_{S⊆N,N◁ G} N.

This is the smallest normal subgroup of G containing S. We shall denote it by Ncl_{G}(S) or by ⟨S^{G}⟩.

The group G with presentation on X with relators and/or relations R (that is, a set of words in the elements of X) is defined to be:

G = F/ Ncl_{F} (R).

where F is the free product on X. For the purposes of taking the normal closure, a relation x = y is regarded as being the relator xy^{−1}. This is usually written as G = ⟨X | R⟩.

2.2.1 Length

Let F be the free product ⋆_{i∈I}A_{i}. (We usually write a free product of A_{1}, …, A_{n} as A_{1} ⋆ … ⋆ A_{n}.) Let w ∈ F be such that w = a_{1}a_{2} … a_{m} when in reduced form. Then we say that w has length m and write l(w) = m. Note that l(1) = 0.

This leaves the problem of how to define length in a free product F on X = {x_{i} : i ∈ I}. We shall use l(w) for the length of w when considered as an element of ⋆_{i∈I}⟨x_{i}⟩ and L(w) for the more usual notion of the length of w. That is, we use L(w) for the length of w when expressed in terms of the x_{i} and x_{i}^{−1}. For example, in the free group on x, y and z and for w = x^{3}y^{−2}zxy^{2}, we have l(w) = 5 whereas L(w) = 9.

Our notion of length of presentation ⟨X | R⟩ is the usual one; that is, the total length (L) of the relators of R. Thus ⟨x, y | x^{2} = y^{4} = (xy)^{5} = (xy^{2})^{5} = 1⟩ has length 31. (There exist variants on the definition of length of a presentation, for example, some people add the number of generators to the length given above.)

2.3 Amalgams—definitions

Definition 2.3.1. Let I be an index set. Then an amalgam is a collection (A, F) of groups A_{J} (I ⊇ J ≠ ∅) and maps f_{JK} : A_{J} ↦ A_{K} (J ⊇ K ⊃ ∅) where f_{JK} is an injective homomorphism (that is, embedding) such that whenever J ⊇ K ⊇ L ⊃ ∅ we have f_{JK}f_{KL} = f_{JL}. (Here, we have our maps acting on the right.) Throughout, we shall only consider the case when I is finite—indeed, I will usually have size 2.

Definition 2.3.2. The universal realisation or universal completion, U, of the amalgam (A, F) is the group generated by A (that is, the A_{J} for J ≠ ∅) subject to the relations af_{JK} = a for all maps f_{JK}(∈ F) and for all a ∈ A_{J}. This defines universal realisation by means of a presentation and so U = F/N where F is the free product of the A_{J} and N is the normal closure of the relators (af_{JK})a^{−1}. Thus, for each J there is a (natural) homomorphism ϕ_{J} : A_{J} → U which is the composition of the natural embedding of U in F and the quotient map F → U.

Definition 2.3.3. We say that G is a realisation or image of the amalgam (A, F) if there is a homomorphism ϕ : U → G that is surjective, that is, any homomorphic image of U is a realisation of the amalgam (A, F). We say that G is a true realisation or true image of the amalgam (A, F) if, for all J, the composition (ϕ_{J}ϕ) : A_{J} → G is faithful on A_{J}.

Definition 2.3.4. Two amalgams (A, F) and (A′, F′) (with the same index set I) are said to be isomorphic or strongly isomorphic if for all J there exist isomorphisms ψ_{J} : A_{J} → A′_{J} such that ψ_{J}f′_{JK} = f_{JK}ψ_{K}. The amalgams (A, F) and (A′, F′) are said to be similar or weakly isomorphic if there exists a permutation σ of I such that for all J there exist isomorphisms ψ̄_{J} : A_{Jσ} → A′_{J} such that ψ̄_{J}f′_{JK} = f_{JσKσ}ψ̄_{K}.

Definition 2.3.5. The free amalgamated product of {A_{i}}_{i∈I} over C (with respect to embeddings f_{i} : C → A_{i}) is the universal realisation of the amalgam (A, F) defined as follows:

(i) A_{i} = A_{i} and A_{J} = C whenever |J| ⩾ 2.
(ii) f_{JK} = 1 whenever |J|, |K| ⩾ 2 or J = K = {i}.
and (iii) f_{J,{i}} = f_{i} whenever J ⊃ {i} and |J| ⩾ 2.

We denote this by (⋆_{i∈I}A_{i})_{C}.

A few remarks are apposite at this juncture. Amalgams that are isomorphic or similar give rise to isomorphic universal realisations (as groups). We shall not apply the terms ‘strongly isomorphic’ and ‘weakly isomorphic’ to amalgams. In general, the universal realisation of an amalgam need not be a true image of it—indeed, it is possible that the universal realisation could be trivial whereas the members of A are not. We note that the free amalgamated product (⋆_{i∈I}A_{i})_{C} is always a true realisation of its amalgam.

2.3.1 The case when we amalgamate two groups

With respect to the notation introduced above, we are considering the case I = {1, 2} and the amalgam ({A_{{1}}, A_{{2}}, A_{{1,2}}}, {f_{{1,2},{1}}, f_{{1,2},{2}}}). We shall use A, B, C, ϕ and θ instead of A_{{1}}, A_{{2}}, A_{{1,2}}, f_{{1,2},{1}}, f_{{1,2},{2}} respectively. We shall write (A, B, C, ϕ, θ) instead of ({A, B, C}, {ϕ, θ}). The general definitions above are reproduced below for the special case of amalgamating two groups. Note that there is not an exact correspondence between the definitions below and those listed previously.

Definition 2.3.6. An amalgam (of groups A and B over their intersection C) is an ordered quintuple (A, B, C, ϕ, θ), where A, B and C are groups and ϕ : C → A and θ : C → B are faithful homomorphisms (or embeddings).

Definition 2.3.7. The universal realisation or universal completion, U, of (A, B, C, ϕ, θ) is the free product A ⋆ B quotiented out by the relations cϕ = cθ for all c ∈ C. This is also called the free amalgamated product of A and B over C (with respect to ϕ and θ) and is denoted by A ⋆_{C} B. We note that A ⋆_{C} B is not independent of ϕ and θ.

Definition 2.3.8. We say that amalgams (A_{1}, B_{1}, C_{1}, ϕ_{1}, θ_{1}) and (A_{2}, B_{2}, C_{2}, ϕ_{2}, θ_{2}) are isomorphic if there exist isomorphisms f_{A} : A_{1} → A_{2}, f_{B} : B_{1} → B_{2} and f_{C} : C_{1} → C_{2} such that for all c ∈ C_{1} we have cϕ_{1}f_{A} = cf_{C}ϕ_{2} and cθ_{1}f_{B} = cf_{C}θ_{2}. We say that the amalgams (A_{1}, B_{1}, C_{1}, ϕ_{1}, θ_{1}) and (A_{2}, B_{2}, C_{2}, ϕ_{2}, θ_{2}) are similar if they are isomorphic or if (B_{1}, A_{1}, C_{1}, θ_{1}, ϕ_{1}) and (A_{2}, B_{2}, C_{2}, ϕ_{2}, θ_{2}) are isomorphic.

We note that for (A_{1}, B_{1}, C_{1}, ϕ_{1}, θ_{1}) and (A_{2}, B_{2}, C_{2}, ϕ_{2}, θ_{2}) to be isomorphic, we must have A_{1} ≅ A_{2}, B_{1} ≅ B_{2} and C_{1} ≅ C_{2}. Thus we shall only consider isomorphism between amalgams (A, B, C, ϕ_{1}, θ_{1}) and (A, B, C, ϕ_{2}, θ_{2}). If the amalgams are similar but not isomorphic, we shall swap A_{1} with B_{1} (and θ_{1} with ϕ_{1}) and assume that the amalgams are isomorphic (with the same conventions applying as before). In picture terms, (A, B, C, ϕ_{1}, θ_{1}) and (A, B, C, ϕ_{2}, θ_{2}) are isomorphic if and only if the following diagram commutes.

[figure: C; A; B; C; A; B; f_{C}; f_{A}; f_{B}; ϕ_{1}; θ_{1}; ϕ_{2}; θ_{2}]

Note that any isomorphism or similarity of amalgams corresponds to an isomorphism between their universal completions. The object of this chapter is to provide a converse to the previous sentence in the case when A and B are finite. In the case A ≅ B there may be (amalgam) similarities which are not isomorphisms. We shall worry about this case when necessary.

2.3.2 Notation

Notation. For groups C ⩽ A, we define Aut_{A}(C) to be N_{Aut(A)}(C)/C_{Aut(A)}(C). We note that Aut_{A}(C) is a subgroup of Aut(C).

2.4 Structure and automorphisms of amalgams

In this section we shall deal only with free amalgamated products. All proofs that have been omitted can be found in Serre [37]. Some of the results of this section look similar to those for free products.

Theorem 2.4.1. Let G be the free amalgamated product (⋆_{i∈I}A_{i})_{C}. For each i we define T_{i} = {1, t_{i,2}, …} to be a right-transversal of C in A_{i} and then we define S_{i} = T_{i} \ {1}. Then each element g ∈ G has a unique (canonical) representation as:

g = a.s_{1}s_{2} … s_{n}

where n ∈ ℕ = {0, 1, 2, …}, a ∈ C and for each m we have s_{m} ∈ S_{i_{m}} such that i_{m} ≠ i_{m+1} for all m.

Proof. See Serre [37], Section 1.2, Theorem 1, Page 3. □

Definition 2.4.2. Let G, S_{i}, etc., be as in the previous theorem. Then the words [a; s_{1}, s_{2}, …, s_{n}] for n ∈ N, a ∈ C and s_{m} ∈ S_{i_{m}} are said to be reduced if and only if for all m we have i_{m} ≠ i_{m+1}. Let g = a.s_{1}s_{2} … s_{n} where [a; s_{1}, s_{2}, …, s_{n}] is reduced. Then we say that g has length n and write l(g) = n. This definition of length is independent of the transversals T_{i} (and S_{i} = T_{i} \ {1}) chosen.

Note. Let G, S_{i}, etc., be as above. Let g = a.s_{1}s_{2} … s_{n} and g̃ = ã.s̃_{1}s̃_{2} … s̃_{m} be reduced. Then gg̃ has length at most n+m and has length exactly n+m if s_{n} and s̃_{1} do not belong to the same S_{i}. If A_{j} ≠ C, then A_{j} ⩽ A_{k}^{h} just if this conjugate is A_{j} itself, so that j = k and h ∈ A_{j}.

Theorem 2.4.3. Every element of finite order in A ⋆_{C} B is conjugate to an element of A ∪ B. Every finite subgroup of A ⋆_{C} B is contained in a conjugate of A or B. (Both statements are true in general free amalgamated products.)

Proof. See Serre [37]. For the first sentence, Section 1.3, Proposition 2 (and Corollary 1), Pages 5–6. For the second sentence, Section 4.3, Theorem 8 (and corollary), Page 36. □

We suspect that the following theorem has been proved before, but we have failed to locate it in the literature.

Theorem 2.4.4. Let A, B and C be finite groups such that A ≠ C ≠ B and let α be an automorphism of A ⋆_{C} B. Then α can be written in the form α = β.γ where β is an inner automorphism and γ permutes the set {A, B}.

Proof. Let G = A ⋆_{C} B and let α ∈ Aut(G). Without loss of generality, we may assume that |A| ⩾ |B|. Then Aα is finite and so conjugate to a subgroup of A or B. Thus, for some g ∈ G, we have (Aα)^{g} = A or B, the latter case occurring only when A and B are isomorphic. For notational convenience, we may suppose that (Aα)^{g} = A in any case.

Now Bα is conjugate to a subgroup of A or B. But if Bα ⩽ A^{h} for some h ∈ G, then B ⩽ (A^{h})α^{−1} = ((Aα)^{gh})α^{−1}, a conjugate of A (since Inn(G) ◁ Aut(G)). This is a contradiction. Thus Bα = B^{h} for some h ∈ G.

To ease the notation further, we may suppose that Aα = A. If Bα = B then we are done, so we may suppose that Bα = B^{h} ≠ B. Now we want to show that (A, B^{h}) is conjugate in G to (A, B) or that G ≠ ⟨A, B^{h}⟩. Now for all a ∈ A and b ∈ B we have (A, B^{bha}) = (A, B^{ha}) = (A, B^{h})^{a}. Therefore, we may assume that

h = a_{1}b_{1}a_{2} … a_{n}b_{n},

for some n ⩾ 1 where a_{i} ∈ A \ C and b_{i} ∈ B \ C may be assumed to be elements of the chosen transversals of A, respectively B, on the cosets of C. Let the canonical form of h^{−1} be

h^{−1} = λβ_{1}α_{1}β_{2}α_{2} … β_{n}α_{n}.

Now C ≅ Cα ⩽ A. Also Cα = (A ∩ B)α = Aα ∩ Bα = A ∩ B^{h}. But if b ∈ B \ C we have that b^{h} = h^{−1}bh has reduced length 2n+1 and so b^{h} ∉ A. Therefore, Cα = A ∩ B^{h} ⩽ A ∩ C^{h}, and by comparison of orders we have Cα = C^{h} ⩽ A. But for c ∈ C, we have that c^{h} ∈ A implies that c^{h} has length 1 or 0. Thus for instance, we must have α_{n}ca_{1} ∈ C. But α_{n} = c̃a_{1}^{−1} for some c̃ ∈ C and so we must have c^{a_{1}} ∈ C for all c ∈ C. Similarly, we must have a_{i}, b_{i} ∈ N_{G}(C) for all i. Therefore, we have shown that Cα = C^{h} = C.

Let k ∈ G have canonical form

k = c̄ā_{1}b̄_{1}ā_{2}b̄_{2} … ā_{m}b̄_{m},

for some m where we may have that ā_{1} = 1 and/or b̄_{m} = 1. Then kα, written in canonical form, has length:

#non-trivial a_{i}’s + (2n+1)#non-trivial b_{i}’s.

Therefore kα ∈ A ∪ B just when k ∈ A in which case kα ∈ A. Thus we have shown that for b ∈ B \ C we have b ∉ Gα. This completes the proof. □

Remark. The previous theorem is not true if one (or both) of A or B is infinite. For example we can take A = ⟨a⟩ ≅ C_{2} and B = ⟨b⟩ ≅ ℤ. Then α : (a, b) ↦ (a, ab) is an automorphism of the free product A ⋆ B, but cannot be expressed in the form α = β.γ as detailed in Theorem 2.4.4. Also, the previous theorem cannot be generalised to three or more factors. Consider, for example, the free product of A = ⟨a⟩ ≅ 2, B = ⟨b⟩ ≅ 2 and C = ⟨c⟩ ≅ 2 and the automorphism (a, b, c) ↦ (a, b^{a}, c).

2.5 When are two amalgams isomorphic?

This entire section is devoted to the proof of the main theorem which we shall state below. The assumption in the theorem is justified by the lemma which precedes it.

Lemma 2.5.1. If the amalgams (A, B, C, ϕ_{1}, θ_{1}) and (A, B, C, ϕ_{2}, θ_{2}) are isomorphic then we must have that Cϕ_{1} and Cϕ_{2} are conjugate in Aut(A) and Cθ_{1} and Cθ_{2} are conjugate in Aut(B). Thus there is no harm in identifying the sets C, Cϕ_{1}, Cϕ_{2}, Cθ_{1} and Cθ_{2}.

Proof. For the first sentence, we have that cϕ_{1}f_{A} = cf_{C}ϕ_{2} and cθ_{1}f_{B} = cf_{C}θ_{2} for all c ∈ C and so Cϕ_{1}f_{A} = Cf_{C}ϕ_{2} = Cϕ_{2} and Cθ_{1}f_{B} = Cf_{C}θ_{2} = Cθ_{2} where f_{A} ∈ Aut(A) and f_{B} ∈ Aut(B). To show that we may identify Cϕ_{1} and Cϕ_{2} we write f_{A} as f′_{A}f′′_{A} where f′_{A}, f′′_{A} ∈ Aut(A) and Cϕ_{1}f′_{A} = Cϕ_{2}. Thus if we replace ϕ_{1} by ϕ_{1}f′_{A} then the statement, “… existence of f_{A} such that … ”, becomes “… existence of f′′_{A} such that … ”. Thus, by a suitable change of notation, we may assume that Cϕ_{1} = Cϕ_{2}. Symmetrical arguments in B show that we may assume that Cθ_{1} = Cθ_{2}. This completes the proof of the second sentence and of the lemma. □

Theorem 2.5.2. The isomorphism classes of (A, B, C, ϕ, θ), with specified images Cϕ ⩽ A and Cθ ⩽ B, are in one-to-one correspondence with the (Aut_{A}(C), Aut_{B}(C))-double cosets of Aut(C) in a very natural way. Thus the number of (isomorphism classes of) amalgams A ⋆_{C} B is the number of (Aut_{A}(C), Aut_{B}(C))-double cosets in Aut(C).

Proof. Let (A, B, C, ϕ, θ) and (A, B, C, ϕ′, θ′) be amalgams with C = Cϕ = Cϕ′ = Cθ = Cθ′. Then these amalgams are isomorphic if and only if there exist f_{A}, f_{B} and f_{C} in Aut(A), Aut(B) and Aut(C) respectively such that for all c ∈ C we have cϕf_{A} = cf_{C}ϕ′ and cθf_{B} = cf_{C}θ′. But the maps f_{C}, ϕ, ϕ′, θ and θ′ can be considered to be automorphisms of C and so we must have f_{A} ∈ N_{Aut(A)}(C) and f_{B} ∈ N_{Aut(B)}(C). Moreover, since we are now considering restrictions to C, we may consider f_{A} and f_{B} to be elements of Aut_{A}(C) and Aut_{B}(C) respectively.

Thus, (A, B, C, ϕ, θ) and (A, B, C, ϕ′, θ′) are isomorphic if and only if there are elements f_{A} ∈ Aut_{A}(C), f_{B} ∈ Aut_{B}(C) and f_{C} ∈ Aut(C) such that for all c ∈ C we have:

[cf_{C} =]cϕf_{A}(ϕ′)^{−1} = cθf_{B}(θ′)^{−1},

where the [cf_{C} =] condition is unnecessary. Replacing cϕ by c and postmultiplying by θ′f_{B}^{−1} we find that the condition becomes that for all c ∈ C there exist f_{A} ∈ Aut_{A}(C) and f_{B} ∈ Aut_{B}(C) such that:

cf_{A}(ϕ′)^{−1}θ′f_{B}^{−1} = cϕ^{−1}θ,

which is true if and only if ϕ^{−1}θ and (ϕ′)^{−1}θ′ are in the same (Aut_{A}(C), Aut_{B}(C))-double coset of Aut(C). □

2.5.1 A useful result

Lemma 2.5.3. Suppose two free amalgamated products A_{1} ⋆_{C_{1}} B_{1} and A_{2} ⋆_{C_{2}} B_{2}, where A_{i} and B_{i} are finite, where A_{i} ≠ C_{i} ≠ B_{i}, are isomorphic as groups. Then, swapping A_{2} and B_{2} if necessary, we have A_{1} ≅ A_{2}, B_{1} ≅ B_{2} and C_{1} ≅ C_{2} with the amalgams corresponding to A_{1} ⋆_{C_{1}} B_{1} and A_{2} ⋆_{C_{2}} B_{2} being isomorphic.

Proof. Let ψ : A_{1} ⋆_{C_{1}} B_{1} → A_{2} ⋆_{C_{2}} B_{2} be a group isomorphism. Without loss, we may assume that |A_{i}| ⩾ |B_{i}|. Now, by Theorem 2.4.3, A_{1}ψ is conjugate to a subgroup of A_{2} or B_{2}. In particular, |A_{1}| = |A_{1}ψ| ⩽ |A_{2}|. Similarly, we have |A_{2}| = |A_{2}ψ^{−1}| ⩽ |A_{1}| and so |A_{1}| = |A_{2}|. So either |A_{2}| > |B_{2}| in which case ψ(A_{1}) is conjugate to A_{2}, or |A_{2}| = |B_{2}| and without loss of generality we have that A_{1}ψ is conjugate to A_{2}. Thus, after conjugation, we may assume that A_{1}ψ = A_{2}. In particular, A_{1} ≅ A_{2}. Now B_{1}ψ is not conjugate a subgroup of A_{2}, since B_{1} = B_{1}ψψ^{−1} is not conjugate to a subgroup of A_{1} = A_{2}ψ^{−1}. So B_{1}ψ is conjugate a subgroup of B_{2} and B_{2}ψ^{−1} is conjugate to a subgroup of B_{1}. Thus |B_{1}| = |B_{2}|, so B_{1}ψ is conjugate to B_{2} and so B_{1} ≅ B_{2}. Now the proof of Theorem 2.4.4 shows that (A_{2}, B_{2}) and (A_{2}, B_{1}ψ) = (A_{1}ψ, B_{1}ψ) are conjugate. Thus we may also assume that B_{1}ψ = B_{2}. Then C_{1}ψ = (A_{1} ∩ B_{1})ψ = A_{1}ψ ∩ B_{1}ψ = A_{2} ∩ B_{2} = C_{2}. Now the restrictions of ψ (to A_{1}, B_{1} and C_{1}) give the maps necessary to show that the amalgams corresponding to A_{1} ⋆_{C_{1}} B_{1} and A_{2} ⋆_{C_{2}} B_{2} are isomorphic. □

Corollary 2.5.4. Two free amalgamated products A⋆_{C}B, where A and B are finite, are isomorphic as groups just when their corresponding amalgams are isomorphic or similar. Thus the number of group-isomorphism classes of amalgams A ⋆_{C} B is the number of (Aut_{A}(C), Aut_{B}(C))-double cosets in Aut(C), except if A ≅ B when this number is merely an upper bound.

Proof. The first sentence follows from the previous lemma. The second sentence is an immediate consequence of Theorem 2.5.2. □

2.6 Some examples

2.6.1 The case of amalgamating two S_{n}’s over S_{n−1}

For n ⩾ 2, we consider the amalgam(s) A ⋆_{C} B where A ≅ B ≅ S_{n} and C ≅ S_{n−1}. If n = 2 or 3 then Aut(C) = Aut_{A}(C) = Aut_{B}(C) = 1 and so the amalgam is unique. If n ⩾ 4 and n ≠ 7 then Aut(C) = Inn(C) ≅ C and so Aut_{A}(C) = Aut_{B}(C) = Aut(C) and the amalgam is unique. If n = 7 then Aut_{A}(C) = Aut_{B}(C) = Inn(C) ≅ C ≅ S_{6} whereas Aut(C) ≅ S_{6}:2 ≅ PΓL_{2}(9) and thus there are two amalgams when n = 7. It turns out that these amalgams are not even similar and that their universal realisations are non-isomorphic. Indeed, they do not even have the same sets of finite homomorphic images, as we shall see below.

In any image of A ⋆_{C} B, the actions of A (respectively B) on the cosets of C induce (permutation) actions of C on n−1 points. We say that A ⋆_{C} B is symmetric (or of symmetric-type) if these permutation actions of C are isomorphic, that is, the point stabiliser of C in one of the actions also stabilises a point in the other one. This set-up occurs, for example, in the symmetric group G = S_{n+1}, with A = G_{α}, B = G_{β} (α ≠ β) and C = G_{αβ} (point-wise). If n ≠ 7, then C has just one action on n−1 points and so the amalgam A ⋆_{C} B must be symmetric.

However, if n = 7 then C has two actions on n−1 = 6 points. The case of A ⋆_{C} B where a point stabiliser in one action (of C on 6 points) stabilises none in the other is said to be anti-symmetric or skewed. The smallest group that arises in this manner is U_{3}(5):2 ≅ PΣU_{3}(5) in which the point stabiliser in the 50 point action is S_{7} with orbits of sizes 1 + 7 + 42. A detailed knowledge of the Hoffman–Singleton graph, see Appendix C, will facilitate understanding of this situation. We note that the groups S_{8} and U_{3}(5):2 are homomorphic images of unique (and non-isomorphic) amalgams of type S_{7} ⋆_{S_{6}} S_{7} (and in a unique way). Another point of note is the following. The images mentioned here all have an (inner) automorphism swapping A and B. In the symmetric case, this must act as an inner automorphism of C and can be chosen to commute with C, but in the skewed case this automorphism must act as an outer automorphism of C and hence extend C to S_{6}:2 ≅ M_{10}:2.

2.6.2 Amalgamating two M_{12}’s over L_{2}(11)

For any L_{2}(11) ⩽ M_{12} we have N_{Aut(M_{12})}(L_{2}(11)) ≅ L_{2}(11):2 ≅ Aut(L_{2}(11)). Thus if A ≅ B ≅ M_{12} are the two groups we are amalgamating and C ≅ L_{2}(11) is their intersection we find that Aut_{A}(C) = Aut_{B}(C) = Aut(C). Thus our ‘formula’ (Corollary 2.5.4) gives just one amalgam M_{12} ⋆_{L_{2}(11)} M_{12}. However, Aut(M_{12}) has two conjugacy classes of L_{2}(11); one type is maximal in M_{12} whereas the other type is not. This leads to three isomorphism classes of M_{12} ⋆_{L_{2}(11)} M_{12} depending on how many of the M_{12}’s the L_{2}(11) is maximal in.

2.6.3 Amalgamating two S_{4}’s over D_{8}

Let A ≅ B ≅ S_{4} and C ≅ D_{8} with C ⩽ A and C ⩽ B. Furthermore, let V_{A} and V_{B} be the normal V_{4}’s of A and B respectively. Then Aut_{A}(C) = Aut_{B}(C) = Inn(C) ≅ 2^{2} which has index 2 in Aut(C) ≅ D_{8} and so our formula gives two isomorphism classes of A ⋆_{C} B. These cases can be distinguished according to whether V_{A} and V_{B} do or do not coincide. If V_{A} = V_{B} then A ⋆_{C} B can be written as 2^{2}:(S_{3} ⋆_{2} S_{3}) and so we reserve the notation S_{4} ⋆_{D_{8}} S_{4} for the case when V_{A} ≠ V_{B}.

The next two examples are more relevant to our work. The first of these examples asserts that a progenitor T^{⋆n}:N in which N simply permutes the T_{i} is uniquely determined (given T, N and an action of N on n points). The second example asserts there are at most φ(m) isomorphism classes of progenitors T^{⋆n}:N in which |N_{N}(T_{i})/C_{N}(T_{i})| = m. We refer the reader to Section 1.2 or Chapter 3 for a definition of a progenitor and for the meaning of the accompanying notation.

2.6.4 Amalgamating N and H × T over H

Such an amalgam is obviously unique. Now H × T has at least an Aut(H) × Aut(T) of automorphisms (and these are just the ones normalising both H and T) of which Aut(H) acts faithfully on H. Thus we have Aut_{B}(C) = Aut(C) where A = N, B = H × T and C = H. This observation will prove useful later.

2.6.5 Amalgamating N and (H × T).m over H.m

Let L = H.m and let K = (H × T).m where C_{K}(T) = H. Then L = ⟨H, μ⟩ where μ^{m} ∈ H. We shall further assume that H is characteristic in L (and condition which nearly always holds in our work). Let ϕ be an automorphism of L that fixes Hμ. Then ϕ extends to an automorphism θ of K. Thus Aut_{K}(L) has index at most φ(m) in Aut(L) where φ is the Euler totient function. Thus there are at most φ(m) amalgams of the form N ⋆_{H.m} (H × T).m.

Progenitors

3.1 Formal definitions

Definition 3.1.1. Let n ⩾ 2 be an integer and let T ≠ 1 be a group. A progenitor is a semi-direct product of the free product of n groups T_{0}, …, T_{n−1}, each isomorphic to T, with N, where N permutes the T_{i} transitively. (The restrictions n ≠ 1 and T ≠ 1 are to ensure non-triviality.) Thus, a progenitor, P say, has the form:

P = T^{⋆n}:N,

where T^{⋆n} denotes the free product of n groups isomorphic to T (≠ 1) and the restrictions on N are as above. Note that we shall usually use G to denote a progenitor and its homomorphic images.

Definition 3.1.2. In the above definition, N is the control subgroup of the progenitor (and any of its homomorphic images).

Definition 3.1.3. Let ϕ : G = T^{⋆n}:N → Ḡ be a homomorphism of G = T^{⋆n}:N onto Ḡ. (So ϕ is an epimorphism.) We say that Ḡ is a true image of G (with respect to ϕ) if ϕ(N) ≅ N, ϕ(T_{i}) ≅ T_{i} ≅ T and ϕ(T_{0}), ϕ(T_{1}), …, ϕ(T_{n−1}) are all distinct. We note that in some definitions of true image, the irredundant condition on the distinctness of ϕ(T_{0}), ϕ(T_{1}), …, ϕ(T_{n−1}) has been omitted.

Definition 3.1.4. We define the subgroups N^{i} and N_{i} of N as shown below:

\begin{matrix}N^{i} = & C_{N}(T_{i}), \\ and N_{i} = & N_{N}(T_{i}). \end{matrix}

Note that the transitivity condition of Definition 3.1.1 implies that the isomorphism classes of the groups N^{i} and N_{i} are independent of i and that N_{i} has index n in N. Note also that for all i we have N^{i} ⩽ N_{i}. See Section 3.3.2 for a generalisation of this notation.

Definition 3.1.5. We say that the progenitor P = T^{⋆n}:N is a permutation progenitor if N^{i} = N_{i} for all i. (This condition is equivalent to N^{i} = N_{i} for some i.)

Definition 3.1.6. All progenitors are monomial progenitors. A proper monomial progenitor is a progenitor which is not a permutation progenitor.

Notation. We usually reserve the notation T^{⋆n}:N for a permutation progenitor, a proper monomial progenitor being indicated by the notation T^{⋆n}:_{m} N.

Definition 3.1.7. A progenitor is said to be a quasi-monomial progenitor if there exists H ◁ N such that N_{i} ∩ H = N^{i} and N/H ≅ N_{i}/N^{i} for all i. Otherwise the progenitor is said to be a true-monomial progenitor.

Definition 3.1.8. Let T be cyclic or elementary abelian. Suppose there exists t_{0} ∈ T_{0} such that ⟨t_{0}^{N_{0}}⟩ = T_{0}. Then we define a symmetric generator to be any conjugate of t_{0} under N. (By t_{0}^{N_{0}}, we mean the set of conjugates of t_{0} under N_{0}.)

3.2 Remarks

We will now make some remarks on the definitions of the previous section. Two more substantial (or important) remarks have been given their own (sub-)sections.

1. We prefer N to act primitively on {T_{0}, …, T_{n−1}}. In the past (see Curtis [13]), it was required that N act 2-transitively on {T_{0}, …, T_{n−1}}. The most common exception to our preference for primitivity is the ‘doubling up’ of symmetric generators to give a progenitor T^{⋆(n+n)}:N.2 (from a progenitor T^{⋆n}:N). It appears that the more interesting cases occur when the orders of T and N are not coprime.

2. Let C = C_{N}(T^{⋆n}). Then N/C acts as a group of monomial transformations on the T^{⋆n} and we have N/C ⩽ M ≅ Aut(T) ≀ S_{n}, see Section 3.2.1 below. Note also that N/C is a group of outer automorphisms of the T^{⋆n}. Usually, we have C = 1 but it is sometimes useful to allow C ≠ 1.

3. A progenitor in which T ≅ 2, the cyclic group of order 2, is necessarily a permutation progenitor.

4. All permutation progenitors are quasi-monomial (with our definition). Also, the subgroup H of Definition 3.1.7 is transitive.

5. The definition of symmetric generators given above does not always specify them up to automorphisms of the progenitor.

6. In this thesis, the T_{i} are either cyclic or elementary abelian. In fact, they are usually cyclic. In general theorems, the T_{i} will be assumed to be arbitrary non-trivial finite groups unless otherwise stated. (The definition of a progenitor does not preclude the T_{i} being infinite, but some of our theorems would not apply in that case.)

3.2.1 A note on monomial groups

We commence this section with an observation.

Lemma 3.2.1. For any group T, the group of automorphisms of T^{⋆n} preserving the set {T_{0}, …, T_{n−1}} is Aut(T)^{n}:S_{n} ≅ Aut(T) ≀ S_{n}.

This is fairly clear, for the group of automorphisms of T^{⋆n} preserving each T_{i} is Aut(T)^{n}. The S_{n} then acts as a coordinate permutation group on (the subscripts of) the T_{i}. Thus, we have:

Definition 3.2.2. The full monomial group of (monomial) automorphisms of T^{⋆n} (which preserves T̄ = {T_{0}, …, T_{n−1}}) is Aut(T) ≀ S_{n} which we denote by M = M(T, n).

3.2.2 Progenitors as amalgams

A progenitor G = T^{⋆n}:N is obviously generated by the subgroups T^{⋆n} and N. Since all the T_{i} are images under the conjugation action of N of a particular T, T_{0} say, G is also generated by N and T_{0}. Thus G is generated by subgroups N and T_{0}:N_{0} intersecting in N_{0}. So let A = N, B = T_{0}:N_{0} and C = N_{0}. Then G is a homomorphic image of the free amalgamated product A ⋆_{C} B. To see that G = A ⋆_{C} B, note that all relations of G hold in A ⋆_{C} B. In particular, we see that T_{0} has just n images under the conjugation action of A = N, where n = |A : C| = |N : N_{0}|.

It is often useful to look at progenitors from this point of view, for example, when writing down a presentation of the progenitor. This viewpoint is also useful when considering representations of the progenitor and for determining whether a given group (or class of groups) is an image (of the progenitor).

3.3 More definitions and notation

3.3.1 Length

From the definition of a progenitor G = T^{⋆n}:N, it follows that any element g ∈ G has a unique representation as:

g = πτ_{1}τ_{2} … τ_{m},

for some m where π ∈ N, τ_{i} ∈ T_{a_{i}} \ {1} for some {a_{i}} such that a_{i} ≠ a_{i+1} for all i. Such a form is called the reduced form of g. The length l(g) of g is defined to be m. It is the same as the length l(w) of w in the free product T^{⋆n}. Elements of N are deemed to have length 0.

A slightly different notion of length, which we denote as L(g), defines the length of g to be the length of the shortest possible representation of w in terms of the symmetric generators and their inverses. (Of course, L(g) is defined only when we have a notion of a symmetric generator.) In all cases we must have that l(g) ⩽ L(g) (when the latter is defined). The latter is akin to the definition of L(w) in Section 2.2.1. We have made no attempt to define a length that would be suitable for the progenitor-as-an-amalgam context.

Similar notions of length apply to N-cosets and (N, N)-double cosets so that l(Ng) = l(Nw) = l(w) and l(NgN) = l(NwN) = l(w) where g = πw and l(w) are as above.

3.3.2 N^{ij}, N_{ij} and related notation

For i and j distinct, we define the groups N^{ij} and N_{ij} by

N^{ij} = N^{i} ∩ N^{j} = C_{N}(⟨t_{i}, t_{j}⟩),

and

N_{ij} = N_{i} ∩ N_{j},

respectively. This notation extends to three or more super/subscripts and thus we define:

N^{i_{1}i_{2}...i_{m}} = N^{i_{1}} ∩ N^{i_{2}} ∩ … ∩ N^{i_{m}} = C_{N}(⟨t_{i_{1}}, t_{i_{2}}, …, t_{i_{m}}⟩),

and

N_{i_{1}i_{2}...i_{m}} = N_{i_{1}} ∩ N_{i_{2}} ∩ … ∩ N_{i_{m}},

where i_{1}, i_{2}, …, i_{m} are distinct. We note that the subgroup N_{i_{1}i_{2}...i_{m}} is just the pointwise stabiliser in N of i_{1}, i_{2}, …, i_{m} where the action of N on the points 0, 1, …, n−1 is induced from its permutation action on the T_{i}.

3.3.3 The coset stabiliser, N^{(w)}

For g ∈ G = T^{⋆n}:N, the coset stabiliser N^{(g)} is defined to be

N^{(g)} = {σ ∈ N | Ng = Ngσ}.

If g = πw where π ∈ N and w ∈ T^{⋆n} then it follows that N^{(g)} = N^{(w)} since Ng = Nπw = Nw. Note also that Ngσ = Ng^{σ} whenever σ ∈ N.

If w involves only elements from T_{i_{1}}, T_{i_{2}}, …, T_{i_{m}} then N^{i_{1}i_{2}...i_{m}} ⩽ N^{(w)}. For example N^{ij} ⩽ N^{(t_{i}t_{j})} and N^{ijk} ⩽ N^{(t_{i}t_{j}^{−1}t_{i}^{−1}t_{k})}. Utilising the notation of Section 3.3.4 below, the inequalities in the last sentence will usually be denoted as N^{ij} ⩽ N^{(ij)} and N^{ijk} ⩽ N^{(ij̄īk)} respectively.

Note that the number of (right-)cosets in NwN is precisely |N|/|N^{(w)}|, since:

\begin{matrix}Nwπ_{1} ≠ Nwπ_{2} ⟺ & Nwπ_{1}π_{2}^{−1} ≠ Nw \\ ⟺ & Nwπ_{1}π_{2}^{−1} ≠ Nw \\ ⟺ & Nw^{π_{1}π_{2}^{−1}} ≠ Nw \\ ⟺ & π_{1}π_{2}^{−1} ∉ N^{(w)} \\ ⟺ & N^{(w)}π_{1} ≠ N^{(w)}π_{2}, \end{matrix}

(see [19] for example). Note also, that if N^{(w)} is transitive (on the T_{i}), then no cosets Nwt_{i} are strictly longer than Nw—except if Nw = N. The concept of a coset stabiliser comes into its own when carrying out manual double coset enumeration. For some examples of this see Section 4.1.

3.3.4 Notation for symmetric generators and other objects

In the case when T_{i} = ⟨t_{i}⟩ is cyclic, we use i to denote t_{i} and īto denote t_{i}^{−1}. Unfortunately, there are two ambiguities associated with this notation which we must resolve if it is to be useful. Firstly, the notation for t_{1} coincides with our usual notation for the identity, so that when this notation is in force, we denote the identity by ⋆ or e. The second ambiguity is 0^{2} which could mean t_{0}^{2} or t_{0}^{t_{2}}. If o(t_{i}) ⩽ 3 then t_{0}^{2} can sensibly be denoted by ⋆ or ¯0 (depending on the order of t_{i}), so that 0^{2} always means t_{0}^{t_{2}} here. However, if t_{i} has order at least 4, then t_{0}^{2} has no such sensible notation and in this case we have not decided on a convention for the meaning of 0^{2}. Thus we shall state the convention we are employing when we use the notation in this case.

If T_{i} ≅ 2^{2} then we can write T_{i} = {1, t_{i}, u_{i}, v_{i}}. In this case we employ the notation i = t_{i}, ī = u_{i} and ¯ ī = v_{i}.

Occasionally, we may stretch the notation g (g ∈ G) so that it denotes the coset Ng as well as the element g. We use [[g]] for the double coset NgN. In particular, note the use of ⋆ for N and [[⋆]] for NeN = N.

3.3.5 Notation for equivalence relations

Equality has its usual meaning, and when used in the context g = h it means that the group elements g and h are equal and not just that the cosets Ng and Nh are equal. We use g ∼ h to mean that Ng = Nh. Thus we do not denote Ng by g when considering equality between cosets.

We have also used the symbol ∼ in another context. Namely, we write g_{i} ∼ h_{i} if there is a group isomorphism ϕ : G = ⟨g_{1}, g_{2}, …, g_{m}⟩ → H = ⟨h_{1}, h_{2}, …, h_{m}⟩ such that g_{i}ϕ = h_{i} for all i.

Ostensibly, we use g ≈ h to mean that g and h have the same order. However, an examination of the typical use of this notation as shown, for example, in Lemma 3.3.1 below suggests that it has a more precise meaning, namely that (we can show that) g is conjugate to h or h^{−1} in G (the ‘universal’ group we are considering) and all its images. We have restricted the use of this notation to groups defined by a presentation or a symmetric presentation (see Section 3.4).

Lemma 3.3.1. In the triangle group ⟨x, y | x^{2} = y^{3} = (xy)^{7} = 1⟩ ≅ (2, 3, 7), the elements [y, x] and xyxyxy^{−1} have the same order.

Proof. We have:

xyxyxy^{−1} = y^{−1}xy^{−1}xy^{−1}xy^{−1}xy^{−1}x.xy^{−1}  (since (xy)^{7} = 1)
= y^{−1}xy^{−1}xy^{−1}xy^{−1}xy  (collecting terms)
= (yxy)^{−1}xy^{−1}xy.yxy  (rewriting)
≈ xy^{−1}xy  (by conjugation)
≈ y^{−1}xyx = [y, x]  (by inversion or conjugation)

Hence the result holds. Throughout we have assumed implicitly that x has order 2 and y has order 3. Note also that [y, x], [x, y] = xy^{−1}xy and xyxy^{−1} have the same order. □

3.4 Extra relations

We have already seen in Section 1.3 that a symmetric presentation takes the form:

Image ≅ \frac{T^{⋆n}:N}{π_{1}w_{1} = π_{2}w_{2} = ⋯ = π_{m}w_{m} = 1}.

Such a presentation has length l(w_{1})+l(w_{2})+⋯+l(w_{m}). This is the length as a symmetric presentation and is different to and independent of the lengths of any ‘equivalent’ ‘⟨x, y, t⟩- presentations’ the group may have (see Section 3.5).

In practice, the relations we consider are of two types. The first type of relation we consider is of the form:

π = w(T_{0}, T_{1}),

and perhaps, very occasionally, π = w(T_{0}, T_{1}, T_{2}). We note from Curtis’s Lemma (see Lemmas 3.6.1 and 3.6.2 below) that ⟨T_{0}, T_{1}⟩ ∩ N ⩽ C_{N}(N^{01}). This may furnish considerable restrictions on π.

The second type of relation consists of specifying the orders of elements πw where w is a short word in the T_{i}. If l(w) = m then we say that such a relation is an m^{th} order relation and that the order of πw is an m^{th} order parameter. Various degenerate cases are excluded from consideration (if they are spotted), such as πw being conjugate to a proper power of π′w′ or πw being conjugate π′w′ with l(w′) < l(w) or if the order of πw is already known. (The last case would happen if π ∈ N_{i} and w ∈ T_{i} for example.) This type of relation may also be subject to the restrictions imposed by Curtis’s Lemma. In ‘most’ progenitors, we have only considered first order parameters and occasionally second order parameters. Only in the ‘smallest’ progenitors have we considered third, fourth and/or higher order parameters.

3.5 ⟨x, y, t⟩-presentations

It is often useful to be able to convert a symmetric presentation into an ‘equivalent’ presentation of a more familiar type (for example, the ‘⟨x, y, t⟩-presentations’ in the title of this section) so that we can perform coset enumeration using existing packages such as Magma. This is also useful when specifying representations (either permutation or matrix) of a progenitor.

The following is a template for a presentation of a permutation progenitor m^{⋆n}:N:

⟨x, y, t | ⟨x, y⟩ ≅ N, t^{m} = 1, [t, N^{0}] = 1⟩.

For an arbitrary monomial progenitor m^{⋆n}:N we have:

⟨x, y, t | ⟨x, y⟩ ≅ N, t^{m} = 1, ⟨t⟩^{N_{0}} = ⟨t⟩ ⟩,

where ⟨x, y⟩ ∼ N and t ∼ t_{0} throughout. Basically, if ⟨X | 𝒳 ⟩ and ⟨Y | 𝒴 ⟩ are presentations for N and T_{0} respectively, then:

⟨X, Y | 𝒳 , 𝒴 , 𝒵 ⟩,

is a presentation for T^{⋆n}:N, where 𝒵 specifies the normalising action of N_{0} on T_{0}. (That is, 𝒵 consists of the relations y_{i}^{π_{j}} = w_{ij}(y_{k}) where Y = {y_{i}} and the π_{j} generate N_{0}.)

In practice, we can produce better presentations if we do not follow these rules literally. For example, consider the progenitor (2^{2})^{⋆6}:_{m} 3·A_{6}, which we will write as 2^{2⋆6}:_{m} 3·A_{6}. Now ⟨x, y | x^{3} = y^{3} = (xy)^{4} = (xy^{−1})^{5} = 1⟩ is a presentation of N ≅ 3·A_{6}, with N_{0} = ⟨x, x^{y}⟩ ≅ 3 × A_{5} and N^{0} = ⟨xx^{y}, x^{y}x⟩ ≅ A_{5}. Thus we obtain:

⟨x, y, t, u | x^{3} = y^{3} = (xy)^{4} = (xy^{−1})^{5} = 1 = t^{2} = u^{2} = [t, u] = t^{x}u = u^{x}tu = t^{x^{y}}tu = u^{x^{y}}t⟩

as a presentation for 2^{2⋆6}:_{m} 3·A_{6}. But ⟨x, t | x^{3} = t^{2} = (xt)^{3} = 1⟩ ≅ A_{4}. Thus:

⟨x, y, t | x^{3} = y^{3} = (xy)^{4} = (xy^{−1})^{5} = 1 = t^{2} = (xt)^{3} = [t, x^{y}x] = [t, xx^{y}]⟩

is a neater presentation for 2^{2⋆6}:_{m} 3·A_{6}. Here, the relations [t, x^{y}x] = [t, xx^{y}] = 1 ensure that t has (at most) 18 images under N and 3 images under N_{0}; the relation (xt)^{3} = 1 ensures that the images of t under N_{0} generate a copy of 2^{2}. Thus T_{0} = {1, t, t^{x}, t^{x^{2}}} ≅ 2^{2} has just 6 images under the conjugation action of N.

3.6 General results

3.6.1 The famous lemma

The following lemmas, although easy, are quite powerful, especially in permutation progenitors. They were both discovered by R. T. Curtis.

Lemma 3.6.1. We always have:

⟨t_{i}, t_{j}⟩ ∩ N ⩽ C_{N}(N^{ij}).

Proof. Let g ∈ ⟨t_{i}, t_{j}⟩ ∩ N. Then if π ∈ N^{ij} it must centralise g since g ∈ ⟨t_{i}, t_{j}⟩. So g ∈ C_{G}(N^{ij}). But g ∈ N, thus g ∈ C_{N}(N^{ij}), as required. □

Lemma 3.6.2. (Curtis) For all m ⩾ 1 (m ∈ ℤ), we have:

⟨t_{i_{1}}, …, t_{i_{m}}⟩ ∩ N ⩽ C_{N}(N^{i_{1}...i_{m}}).

Proof. Essentially the same as the previous lemma. □

3.6.2 Results for involutory generators

At least one result in this section relies on the fact that a group generated by two involutions is dihedral. Other results rely on the fact that we are dealing with permutation progenitors. Yet more results rely on the simple fact that life is less complicated when we are dealing with involutions. For some results in this section it is essential to have N acting primitively on the symmetric generators and so we shall make such an assumption throughout this section.

Lemma 3.6.3. Let G be an image of a progenitor 2^{⋆n}:N (with N acting primitively on the symmetric generators) such that there exist distinct i and j with t_{i}t_{j} ∈ N. Then N has index at most 2 in G.

Proof. Define a relation ϱ on {0, 1, …, n−1} by i ϱ j if and only if t_{i}t_{j} ∈ N. Now t_{i}t_{i} = 1 ∈ N, so ϱ is reflexive and t_{i}t_{j} ∈ N implies that t_{j}t_{i} = (t_{i}t_{j})^{−1} ∈ N, so ϱ is also symmetric. Moreover, t_{i}t_{j}, t_{j}t_{k} ∈ N implies that t_{i}t_{k} = t_{i}t_{j}.t_{j}t_{k} ∈ N; thus ϱ is transitive. Hence ϱ is an equivalence relation. It is easily seen that ϱ is preserved by N—for all π ∈ N we have t_{iπ}t_{jπ} = (t_{i}t_{j})^{π} ∈ N, so that iπ ϱ jπ whenever i ϱ j. Thus ϱ is a non-trivial N-congruence and hence is the universal relation. □

Lemma 3.6.4. Let G be an image of a progenitor 2^{⋆n}:N (with N acting primitively on the symmetric generators) such that there exist distinct i, j and k with t_{i}t_{j}t_{k} ∈ N. Then G = N or N^{ijk} = N^{ij} = N^{jk}.

Proof. Suppose that N^{ijk} ≠ N^{ij} (which is if and only if N^{ij} ⩽̸ N^{k}), so that there exists π ∈ N^{ij} such that kπ = l ∉ {i, j, k}. Then t_{i}t_{j}t_{k}, t_{i}t_{j}t_{l} ∈ N and so t_{l}t_{k} = (t_{i}t_{j}t_{l})^{−1}t_{i}t_{j}t_{k} ∈ N and the previous lemma and its proof shows that t_{i}t_{j} ∈ N and t_{i} ∼ t_{j} for all i and j. Thus t_{k} ∼ t_{i}t_{j}t_{k} ∼ ⋆ and so t_{k} ∈ N. Therefore t_{i} ∈ N for all i and so G = N.

Moreover, t_{k}t_{j}t_{i} = (t_{i}t_{j}t_{k})^{−1} ∈ N, so by the above paragraph we have N^{ijk} = N^{jk} or G = N. This completes the proof. □

Note. Of course, the previous lemma is trivially true if i, j and k are not distinct.

The next lemma is useful for those cases when N is ‘small’ (for example, sharply 2-transitive) and Curtis’s Lemma is impotent.

Lemma 3.6.5. In any image of 2^{⋆n}:N (with N acting primitively on the symmetric generators) such that |G : N| > 2 we have that any element of ⟨t_{i}, t_{j}⟩ ∩ N has order at most 2. Furthermore, such elements must fix (setwise) the set {i, j} except in a few cases which we comment on in a special section after the proof below. (These cases may not occur.)

Proof. From Lemma 3.6.3, we see that the cosets N, Nt_{0}, …, Nt_{n−1} are distinct. Let π ∈ ⟨t_{i}, t_{j}⟩∩N. Then there are two cases to consider, namely π = [t_{i}t_{j}]^{m} and π = [t_{i}t_{j}]^{m}t_{i} (for some m).

If π = [t_{i}t_{j}]^{m}, then π̃ = [t_{j}t_{i}]^{m} = π^{−1} ∈ ⟨t_{i}, t_{j}⟩ ∩ N. Therefore, t_{i}^{π} ∼ t_{i}π = t_{i}[t_{i}t_{j}]^{m} = [t_{j}t_{i}]^{m}t_{i} = π̃t_{i} ∼ t_{i} and so t_{i}^{π} = t_{i}. Similarly, we find that t_{j}^{π} = t_{j}. Therefore π ∈ Z(⟨t_{i}, t_{j}⟩), i.e., π is central in a dihedral group, and so π has order 2. Also, π as a permutation fixes both i and j.

If π = [t_{i}t_{j}]^{m}t_{i} then π has order 2. Let π̃ = [t_{j}t_{i}]^{m}t_{j}. Then it is the case that π̃ has order 2, but we do not know that π̃ ∈ N. However, if π̃ ∈ N then t_{i}^{π} ∼ t_{i}π = t_{i}[t_{i}t_{j}]^{m}t_{i} = [t_{j}t_{i}]^{m}t_{j}t_{j} = π̃t_{j} ∼ t_{j} and thus t_{i}^{π} = t_{j}. Similarly (or from the fact that π has order 2), we find that t_{j}^{π} = t_{i} and so π as a permutation swaps i and j. Also we have that π = π^{π} = [t_{j}t_{i}]^{m}t_{j} = π̃.

One condition that ensures that π̃ ∈ N is that there is a element g ∈ N that swaps i and j so that ([t_{i}t_{j}]^{m}t_{i})^{g} = [t_{j}t_{i}]^{m}t_{j}, that is, π̃ = π^{g}. □

What are the exceptional N in the previous lemma?

Let N be a group acting primitively on Ω = {0, 1, …, n−1}. This action induces an action of N on Ω × Ω and the number of orbits of N on Ω × Ω, which is equal to the number of orbits of N_{0} on Ω, is called the rank of N. An orbit, O, of N on Ω × Ω is said to be symmetric if it contains (a, b) and (b, a) for some a and b, and antisymmetric if it contains (a, b) but not (b, a) for some a and b. (There do not exist (a_{1}, b_{1}), (a_{2}, b_{2}) ∈ O such that (b_{1}, a_{1}) ∈ O but (b_{2}, a_{2}) ∉ O.)

If N has odd order, then each orbit of N (on Ω × Ω) is antisymmetric except the diagonal orbit {(a, a)}. If N has even order and rank 2 (i.e., is 2-transitive) or rank 3, then each orbit of N is symmetric. Since the exceptions to the lemma occurred when N contained an involution and had an antisymmetric orbit, it follows that N has even order and rank at least 4.

3.6.3 Collapse of N

In this section, we shall consider a progenitor G = T^{⋆n}:N and what happens if we have a homomorphism ϕ : G → Ḡ = Gϕ such that N̄ = Nϕ ≇ N. In this section, the notation will be as in the previous sentence, except that we may have N̄ ≅ N. Furthermore, we shall require N to act primitively on T̄ = {T_{0}, …, T_{n−1}}. Let c(N) = N_{0,1,...,n−1}. Then we also require that Ncl_{T_{i}:⟨g⟩}(⟨g⟩) = T_{i}:⟨g⟩ for all g ∈ c(N) \ {1}.

Many progenitors with we work have N acting primitively on the T_{i} and most of these have c(N) = 1 and so automatically satisfy vacuous condition that Ncl_{T_{i}:⟨g⟩}(⟨g⟩) = T_{i}:⟨g⟩ for all g ∈ c(N) \ {1} and hence they satisfy all the conditions of the above paragraph. There are some progenitors like 3^{⋆4}:_{m} 2·S_{4}^{+} and (2^{2})^{⋆6}:_{m} 3·A_{6} in which N acts primitively on the T_{i} but c(N) ≠ 1. In these cases, we have that Ncl_{T_{i}:⟨g⟩}(⟨g⟩) = T_{i}:⟨g⟩ for all g ∈ c(N) \ {1}, so these progenitors satisfy all the conditions of the above paragraph also.

Theorem 3.6.6. Let G = T^{⋆n}:N be a progenitor satisfying all the conditions of the first of the above two paragraphs. If Ḡ is an image such that N̄ ≇ N then |Ḡ : N̄| ⩽ |T |.

Proof. Let K = ker ϕ ∩ N. Then N̄ ≇ N is equivalent to K ≠ 1 (since N̄ ≅ N/K). Since K ◁ N and N acts primitively on the T_{i} it follows that K acts transitively on the T_{i} or that K ⩽ c(N). In the former case we have that T_{i}ϕ = T_{j}ϕ for all i and j and hence |Ḡ : N̄| ⩽ |T |. In the latter case we have gϕ = 1 for some g ∈ c(N) \ {1} and this forces ϕ(T_{i}:⟨g⟩) = ϕ(Ncl_{T_{i}:⟨g⟩}(⟨g⟩)) = 1. In particular, T_{i}ϕ = 1 for all i and so |Ḡ : N̄| = 1. □

Corollary 3.6.7. Let G, N, Ḡ and N̄ be as above. Then if |Ḡ : N̄| > |T | the permutation action of Ḡ on the cosets of N̄ is faithful.

Proof. Let ψ be the homomorphism ψ : Ḡ → S_{|Ḡ:N̄|} induced by the action of Ḡ on the cosets of N̄. Then N̄ψ has index |Ḡ : N̄| in Ḡψ (since Ḡψ is transitive and N̄ψ is the point stabiliser). Therefore |Ḡψ : N̄ψ| > |T | and so N̄ψ ≅ N̄ ≅ N and so we have ker ψ ∩ N = 1. But ker ψ = CoreḠ(N̄) ⩽ N̄ and so ker ψ = 1. □

Progenitors of the form m^{⋆(n+n)}:N.2

The progenitors which we most often encounter in which N does not act primitively on the T_{i} are those of the form m^{⋆(n+n)}:N.2 where N has two orbits of size n on the T_{i} which are fused in N.2 and moreover N acts primitively each of the orbits. In such a case, a collapse of N may still give rise to a non-trivial homomorphic image of G which is (essentially) an image of m^{⋆2}:2.

However a progenitor 2^{⋆(n+n)}:N.2 essentially has no non-trivial images in which N has collapsed to a proper image since 2^{⋆2}:2 is the infinite dihedral group. Also a progenitor such as 5^{⋆(176+176)}:_{m} 2·HS:2 has no non-trivial images in which N has collapsed to a proper image since the unique minimal normal subgroup of N is generated by an involution which inverts all the symmetric generators.

Note that in general we can make no assertions about collapse of N causing essentially complete collapse of G. For example, the only ‘interesting’ images of 2^{⋆14}:2^{3}·L_{3}(2) that we have found are actually images of 2^{⋆7}:L_{3}(2).

3.7 When are two progenitors isomorphic?

We have already answered this question in Chapter 2, since a progenitor is a free amalgamated product of two finite groups. Thus the lemma below is just a restatement of Corollary 2.5.4 (and Theorem 2.5.2) in the language of progenitors.

Lemma 3.7.1. (Identification) Two progenitors T^{⋆n}:N and S^{⋆m}:M are isomorphic only when N ≅ M and T_{0}:N_{0} ≅ S_{0}:M_{0} or N ≅ S_{0}:M_{0} and T_{0}:N_{0} ≅ M and not always then, the question of isomorphism depending on whether the corresponding amalgams ((N, T_{0}:N_{0}, N_{0}) and (M, S_{0}:M_{0}, M_{0})) are isomorphic or similar. Indeed, the number of isomorphic classes of progenitors T^{⋆n}:N is the number of (Aut_{N}(N_{0}), Aut_{T_{0}:N_{0}}(N_{0}))-double cosets in Aut(N_{0}).

3.8 Automorphisms of the progenitors

We have already seen in Chapter 2 what automorphisms of free amalgamated products. Owing to time constraints, we did not manage to translate this into a classification of automorphisms of progenitors. However, we can divide the outer automorphisms of G = T^{⋆n}:N into three categories as follows:

(i) Out_{1}(G), which consists of those automorphisms that extend N to N_{M(T,n)}(N).

(ii) Out_{2}(G). These automorphisms fix N element-wise and fix set-wise, each of the cosets of N_{0} in T_{0}:N_{0}.

(iii) Out_{3}(G). Out_{3}(G) is usually trivial. If not, then Out_{3}(G) ≅ 2 and we are in the case when N ≅ T_{0}:N_{0} and there is a similarity of the corresponding amalgam.

We conjecture that Out(G) ≅ (Out_{2}(G): Out_{1}(G)): Out_{3}(G).

3.9 How many progenitors?

We merely give an example. We shall recall that a progenitor is a free amalgamated product of N and T_{0}:N_{0} over N_{0}. We must therefore calculate Aut_{N}(N_{0}), Aut_{T_{0}:N_{0}}(N_{0}) and Aut(N_{0}).

3.9.1 The progenitor 7^{⋆15}:_{m} 3·A_{7}

Here, we propose to take N ≅ 3·A_{7}, N_{0} ≅ 3 × L_{3}(2) and T_{0}:N_{0} ≅ 7:3 × L_{3}(2). Then we calculate that Aut_{N}(N_{0}) ≅ L_{3}(2), Aut_{T_{0}:N_{0}}(N_{0}) ≅ L_{3}(2):2 and Aut(N_{0}) ≅ 2 × L_{3}(2):2 and hence that there are two (Aut_{N}(N_{0}), Aut_{T_{0}:N_{0}}(N_{0}))-double cosets of Aut(N_{0}). Thus there are two possible progenitors 7^{⋆15}:_{m} 3·A_{7} up to isomorphism.

3.10 The Campbell and Robertson approach

The phrase ‘symmetric presentation’ is also used in connection with work of Campbell and Roberston. In their approach, we start with a finitely presented group:

G = ⟨x_{1}, x_{2}, …, x_{n} | r_{j}(x_{1}, x_{2}, …, x_{n}) = 1 for all 1 ⩽ j ⩽ m⟩.

Then for a (transitive) permutation group P we define the symmetrically presented group P G by:

P G = ⟨x_{1}, x_{2}, …, x_{n} | r_{j}(x_{1σ}, x_{2σ}, …, x_{nσ}) = 1 for all 1 ⩽ j ⩽ m, σ ∈ P ⟩.

In their approach P is usually either the cyclic group ⟨(1, 2, …, n)⟩ or the full symmetric group S_{n}. In some sense, their approach is related to, but is much more restricted than ours. We illustrate their approach with the following example.

Let G = ⟨x_{1}, x_{2}, x_{3}, x_{4} | x_{1}x_{2} = x_{3}⟩ and let P = ⟨(1, 2, 3, 4)⟩. Then

P G = ⟨x_{1}, x_{2}, x_{3}, x_{4} | x_{1}x_{2} = x_{3}, x_{2}x_{3} = x_{4}, x_{3}x_{4} = x_{1}, x_{4}x_{1} = x_{2}⟩

is cyclic of order 5 and P G is one of a class of groups known as Fibonacci groups.

Some examples

4.1 Manual double coset enumeration—examples

4.1.1 A presentation for PGL_{2}(7)

For our first example, we consider the group

G = \frac{2^{⋆4}:S_{4}}{(2, 3) = t_{0}t_{1}t_{0}t_{1}} ,

that is, the progenitor 2^{⋆4}:S_{4} quotiented out by the relation (2, 3) = t_{0}t_{1}t_{0}t_{1}. A presentation for this group is:

⟨x, y, t | x^{4} = y^{2} = (xy)^{3} = t^{2} = 1 = [t, y] = [t^{x}, y] = y[t, x]^{2}⟩,

where x ∼ (0, 1, 2, 3), y ∼ (2, 3) and t ∼ t_{0}. In the above presentation, all the relations but the last one define the progenitor 2^{⋆4}:S_{4} and the last relation specifies that (2, 3)(t_{0}t_{1})^{2} = 1, which is equivalent to (2, 3) = t_{0}t_{1}t_{0}t_{1}.

We now have to enumerate the double cosets of N = ⟨x, y⟩ ≅ S_{4} in G. It is quite easy to see (using the 4-transitivity of N) that the double cosets of N of length at most 4 in the symmetric generators are:

[[⋆]], [[0]], [[01]], [[010]], [[012]], [[0101]], [[0102]], [[0120]], [[0121]] and [[0123]],

which is shorthand for N = NeN, Nt_{0}N, Nt_{0}t_{1}N and so on.

The easy part of this coset enumeration is to use the relation (2, 3) = t_{0}t_{1}t_{0}t_{1} to show that Nt_{0}t_{1}t_{0} = N(2, 3)t_{1} = Nt_{1}, which we write as 010 ∼ 1. From this, by postmultiplying by t_{0}, t_{1} and t_{2} (respectively), we may also conclude that 01 ∼ 10, 0101 ∼ ⋆ and 0102 ∼ 12. The above yields the double coset equalities: [[010]] = [[0]], [[0101]] = [[⋆]] and [[0102]] = [[01]]. Thus we see that we have coset stabilisers: N^{(⋆)} = N, N^{(0)} ⩾ N^{0} = ⟨(1, 2, 3), (2, 3)⟩ and N^{(01)} ⩾ ⟨N^{01}, (0, 1)⟩ = ⟨(0, 1), (2, 3)⟩. (We have (0, 1) ∈ N^{(01)} because of 01 ∼ 10.)

We now consider the double coset [[012]]. From 01 ∼ 10 and hence 012 ∼ 102, it is plain that N^{(012)} ⩾ ⟨(0, 1)⟩. Moreover, we have 012 = 0.1212.21 = 0(0, 3)21 ∼ 321, that is Nt_{0}t_{1}t_{2} = t_{3}t_{2}t_{1} where t_{3}t_{2}t_{1} = (t_{0}t_{1}t_{2})^{(0,3)(1,2)}. Therefore N^{(012)} ⩾ ⟨(0, 1), (0, 3)(1, 2)⟩ ≅ D_{8}, a transitive subgroup of N. So any coset of length 4 is reducible. In particular, we have 012 ∼ 102 ∼ 320 ∼ 230 ∼ 103 ∼ 013 ∼ 231 ∼ 321 and so 012.0 ∼ 320.0 ∼ 32, 012.1 ∼ 32, 012.2 ∼ 01 and 012.3 ∼ 01. Thus [[0120]] = [[0121]] = [[0123]] = [[01]] and so there are no (irreducible) cosets of length 4.

Thus we have completed our double coset enumeration and shown that the index of N in G is (at most) 14 and that the order of G is at most 336. To show that G really does have order 336, we give generators for PGL_{2}(7) satisfying our presentation of G; namely x ∼ (∞, 4, 6, 1)(0, 5, 2, 3), y ∼ (∞, 0)(1, 6)(2, 3)(4, 5) and t ∼ (1, 6)(2, 5)(3, 4). (These correspond to the linear fractional maps η ↦ \frac{4η+2}{η−1}, η ↦ −\frac{1}{η} and η ↦ −η respectively.) We exhibit the (collapsed) Cayley diagram for this group as shown below:

[figure: 1; 4; 6; 3; [[⋆]]; [[0]]; [[01]]; [[012]]; 4; 3; 2; 1; 2; 4]

Note that, in general, the Cayley graph on cosets of N with joins Nw—Nwt_{i} is not regular, and that includes the above graph, for the cosets Nt_{i}t_{j}t_{k} have valence 2, the cosets Nt_{i}t_{j} have valence 3 and the cosets Nt_{i} and N have valence 4. Such a graph will admit N as a group of automorphisms (usually faithfully), but will not, in general, admit G. (The automorphism group of the above Cayley graph is N ≅ S_{4}.)

However, the graph on the cosets Nw with joins Nw—Nt_{i}w is regular, of valence 4, and admits G ≅ PGL_{2}(7) as a group of automorphisms. In fact, the automorphism group of the regular graph is precisely G. This graph is considered to be associated with the above Cayley graph. This situation is typical, as the following lemma shows.

Lemma 4.1.1. Let G be a homomorphic image of a progenitor m^{⋆n}:N, where N acts faithfully on the symmetric generators. Let the number of symmetric generators be an for some a. Suppose that the an + 1 cosets N and Nt, for t a symmetric generator, are distinct. Define a graph Γ whose |G : N| vertices are the cosets, Nw, of N in G, such that Nw is joined to Ntw just when t is a symmetric generator. Then Γ is a regular graph of valence an with no loops or multiple edges. Moreover, the action of G on Γ defined by (Nw).g = N(wg) induces an embedding of G into the automorphism group of Γ.

Proof. Let Nw = Nw̃. Then w̃w^{−1} ∈ N and so Ntw̃ = Nτw where τ = t^{w̃w^{−1}} is a symmetric generator. This shows that the valence of Γ is at most an. Now Nw = Ntw implies that N = Ntww^{−1} = Nt, a contradiction. Similarly, Ntw = Nτw (for t and τ symmetric generators) implies that Nt = Nτ, a contradiction. This completes the proof of all but the last sentence of the lemma.

It is obvious that Γ admits G as a group of automorphisms. (The edge Nw—Ntw is mapped to the edge Nwg—Ntwg.) To see that G acts faithfully, we note that the action of G on Γ is just the action of G on the cosets of N, so that the kernel of this action is a subgroup of N. But N acts faithfully on the cosets Nt (since these are all distinct). Thus N and G act faithfully on Γ. □

The regular graph turns out to be very well known, for it is related to the order 2 projective plane. In fact, we can label the vertices of the graph with the points and lines of the order 2 projective plane in such a way that the neighbours of a given line are just the 4 points it does not contain. Similarly, the neighbours of a given point are just the 4 lines that do not contain it. The following table:

N 0 Nt_{1} 3 Nt_{0}t_{1} 3 Nt_{2}t_{3} 1 Nt_{0}t_{1}t_{2} 1
Nt_{0} 0 Nt_{2} 6 Nt_{0}t_{2} 6 Nt_{1}t_{3} 2 Nt_{0}t_{2}t_{1} 2
Nt_{3} 5 Nt_{0}t_{3} 5 Nt_{1}t_{2} 4 Nt_{0}t_{3}t_{1} 4

gives a correspondence between the cosets of N and the points (large numerals) and lines (small numerals) of the ‘standard’ (order 2) projective plane. The action of G on the cosets of N gives the 14-point action of G as shown in the table below. For reference, we have also included the 8-point action that we gave earlier.

Element Realisation  8-point  14-point
in 2^{⋆4}:S_{4}  action  action
x  (0, 1, 2, 3) (∞, 4, 6, 1)(0, 5, 2, 3)  (2, 6)(1, 5, 3, 4)(1,4)(0,3,6,5)
y  (2, 3)  (∞, 0)(1, 6)(2, 3)(4, 5)  (2, 4)(5, 6)(2,4)(5,6)
t  t_{0}  (1, 6)(2, 5)(3, 4)  (0, 0)(1, 1)(2, 2)(3, 3)(4, 4)(5, 5)(6, 6)

We refer the reader to Section 7.5 to see a Coxeter presentation corresponding to the symmetric presentation given here. Finally, let us label the cosets N, Nt_{i}, Nt_{i}t_{j} and Nt_{i}t_{j}t_{k} by ⋆, the point i, the unordered pair ij and the partition ij.kl respectively. Then the projective plane referred above, which is also known as the Fano plane, takes the form shown below.

⋆ 01 23 03  02 12 13 0  1 2  3^{01.23}^{02.13}  ^{03.12}

4.1.2 A presentation for PGL_{2}(11)

In this case we consider the group:

G = \frac{2^{⋆6}:L_{2}(5)}{[(0, 2, 4, 1, 3)t_{0}]^{4} = 1}.

The double cosets of N of length at most 3 are:

[[⋆]], [[∞]], [[∞0]], [[∞0∞]], [[∞01]] and [[∞02]].

The above relation is (0, 2, 4, 1, 3)^{4}t_{1}t_{4}t_{2}t_{0} = 1 which gives 024 ∼ 1 from which we obtain [[∞01]] = [[∞]]. Also, from 024 ∼ 1 we obtain ∞0 ∼ 31 = [∞0]^{(∞,3,2)(0,1,4)}, and so N^{(∞0)} ⩾ ⟨(∞, 3, 2)(0, 1, 4), (1, 4)(2, 3)⟩. Thus the cosets [[⋆]], [[∞]] and [[∞0]] contain at most 1, 6 and 10 cosets (of N) respectively. From ∞0 ∼ 31 we obtain ∞03 ∼ 313, thus showing that [[∞02]] = [[∞0∞]]. Now ∞0∞141 = ∞(4, 2, 1, ∞, 0)241 ∼ 0241 ∼ ⋆ and so ∞0∞ ∼ 141 = [∞0∞]^{(∞,1,2)(0,4,3)} and so N^{(∞0∞)} ⩾ ⟨(∞, 1, 2)(0, 4, 3), (1, 4)(2, 3)⟩ ≅ A_{4}, a transitive subgroup of N. This [[∞0∞]] contains at most 5 cosets. Moreover, since all cosets of length 3 are in [[∞0∞]] and N^{(∞0∞)} is transitive, it follows that there are no (new) cosets of length at least 4. This completes the coset enumeration and shows that |G : N| ⩽ 22 and that |G| ⩽ 1 320. The Cayley diagram of this coset enumeration is shown below.

[figure: 1; 6; 10; 5; [[⋆]]; [[∞]]; [[∞0]]; [[∞0∞]]; = [[∞02]]; 6; 5; 3; 1; 3; 6]

As with the PGL_{2}(7) case, the regular graph associated with the above Cayley graph is bipartite, the two parts (each with 11 vertices) being called points and lines, with two vertices being joined just if one of them is a line and the other is a point not on the line. The associated incidence structure is referred to as a biplane since any two points determine two lines and any two lines have just two points in common. The following table shows a correspondence between the cosets of N and the elements of the biplane:

N 0 Nt_{0} 1 Nt_{∞}t_{0} 8 Nt_{0}t_{∞} 1 Nt_{∞}t_{0}t_{∞} 8
Nt_{∞} 0 Nt_{1} 3 Nt_{∞}t_{1} 2 Nt_{1}t_{∞} 3 Nt_{∞}t_{1}t_{∞} 2
Nt_{2} 9 Nt_{∞}t_{2} 6 Nt_{2}t_{∞} 9 Nt_{∞}t_{2}t_{∞} 6
Nt_{3} 5 Nt_{∞}t_{3} 7 Nt_{3}t_{∞} 5 Nt_{∞}t_{3}t_{∞} 7
Nt_{4} 4 Nt_{∞}t_{4} X Nt_{4}t_{∞} 4 Nt_{∞}t_{4}t_{∞} X

where, as in the PGL_{2}(7) case, the large numerals are the points, and the small numerals the lines, of the biplane. (Note that ‘X’ stands for ‘10’.) Thus we see that line 0 contains the points 2, 6, 7, X, 8, line 1 contains the points 1, 5, 6, 9, 7, and so on. Conversely, point 0 is the intersection of lines 2, 6, 7, X, 8, point 1 is the intersection of lines 1, 5, 6, 9, 7, and so on. Thus, with this labelling of the cosets of N, the action of G on the cosets of N gives rise to the permutations on 22 letters as shown below:

x ∼ (1, 3, 9, 5, 4)(2, 6, 7, X, 8)(1,3,9,5,4)(2,6,7,X,8),
y ∼ (1, 4, 2)(3, 8, 9)(5, X, 6)(0,1,3)(2,8,7)(4,5,9),
and t ∼ (0, 0)(1, 1)(2, 2)(3, 3)(4, 4)(5, 5)(6, 6)(7, 7)(8, 8)(9, 9)(X, X),

where x ∼ (0, 1, 2, 3, 4), y ∼ (∞, 0, 1)(2, 4, 3) and t ∼ t_{∞} in the progenitor 2^{⋆6}:L_{2}(5). To show the isomorphism of G with PGL_{2}(11), we realise G as permutations on 12 points as shown below:

\begin{matrix}x ∼ & (1, 3, 9, 5, 4)(2, 6, 7, X, 8), \\ y ∼ & (∞, 6, 7)(0, 3, 9)(1, 8, 5)(2, 4, X), \\ and t ∼ & (1, X)(2, 9)(3, 8)(4, 7)(5, 6). \end{matrix}

These correspond to the linear fractional transformations (of PG_{1}(11) = 𝔽_{11} ∪ {∞}): η ↦ 3η, η ↦ \frac{η+2}{2η−3} and η ↦ −η respectively.

4.1.3 A presentation for L_{2}(19)

In this case, we consider the group:

G = \frac{2^{⋆6}:L_{2}(5)}{[(∞, 0, 1)(2, 4, 3)t_{∞}]^{5} = 1}.

We realise the image L_{2}(19) by letting x, y and t be the linear fractional maps:

η ↦ \frac{η − 5}{5η + 1}, η ↦ \frac{2η}{η + 3} and η ↦ −\frac{1}{η},

respectively, where x ∼ (0, 1, 2, 3, 4), y ∼ (∞, 0, 1)(2, 4, 3) and t ∼ t_{∞} in the progenitor 2^{⋆6}:L_{2}(5). When we proceed to do the double coset enumeration, it is helpful to use an additional relation, so we must show that it is implied by the existing relation.

Lemma 4.1.2. In 2^{⋆6}:L_{2}(5), the relation [(∞, 0)(1, 4)t_{1}]^{5} = 1 (equivalently, t_{1}t_{4}t_{1}t_{4}t_{1} = (∞, 0)(1, 4)) is a consequence of [(∞, 0, 1)(2, 4, 3)t_{∞}]^{5} = 1.

Proof. Note that [(∞, 0, 1)(2, 4, 3)t_{∞}]^{5} = 1 is equivalent to t_{∞}t_{0}t_{1}t_{∞}t_{0} = (∞, 1, 0)(2, 3, 4). Thus we have that:

\begin{matrix}t_{∞}t_{0}.t_{1}t_{4}t_{1}t_{4}t_{1}.t_{∞}t_{0} = & [t_{∞}t_{0}t_{1}t_{∞}t_{0}.t_{0}t_{∞}t_{4}t_{0}t_{∞}]^{2}.t_{∞}t_{0}t_{1}t_{∞}t_{0} \\ = & [(∞, 1, 0)(2, 3, 4).(∞, 0, 4)(1, 2, 3)]^{2}.(∞, 1, 0)(2, 3, 4) \\ = & (∞, 0)(1, 4), \end{matrix}

and so t_{1}t_{4}t_{1}t_{4}t_{1} = t_{0}t_{∞}(∞, 0)(1, 4)t_{0}t_{∞} = (∞, 0)(1, 4) as required. □

Hence, the double coset enumeration proceeds as shown below:

Double #Cosets Coset stabilising Comments
coset  subgroup
[[⋆]]  1  N  —
[[∞]]  6  N^{∞}  —
[[∞0]]  30  N^{∞0}  —
[[∞0∞]]  —  —  This is the double coset [[∞0]] since ∞0∞ ∼
∞0. This follows from [(∞, 0)(1, 4)t_{1}]^{5} = 1.

[[∞01]] — — This is the double coset [[∞0]] since ∞01 ∼ 0∞.

This follows from [(∞, 0, 1)(2, 4, 3)t_{∞}]^{5} = 1.

[[∞02]] 20 ⟨(∞, 2, 3)(0, 4, 1)⟩ The group N^{∞02} is trivial. However, we have ∞02 = ∞(0, 3, 2)(∞, 4, 1)203 ∼ 4203 ∼ 243 = [∞02]^{(∞,2,3)(0,4,1)}, so (∞, 2, 3)(0, 4, 1) ∈ N^{(∞02)}.

[[∞02i]] — — For π ∈ N^{(∞02)} we have ∞02i^{π} = [∞02i]^{π}, so for the purposes of locating the double coset equal to [[∞02i]] it is sufficient to do this when i is an orbit representative under the action of N^{(∞02)}. Now ∞022 = ∞0 so that [[∞02i]] = [[∞0]] for i ∈ {∞, 2, 3}. Also, ∞020 = ∞(∞, 1)(0, 2)02 ∼ 102 so that [[∞02i]] = [[∞02]] for i ∈ {0, 1, 4}.

Thus we have shown that |G : N| = 57 and that |G| = 3 420, hence completing the proof that G ≅ L_{2}(19). We give the Cayley diagram for the coset enumeration below.

[figure: 1; 6; 30; 20; [[⋆]]; [[∞]]; [[∞0]]; [[∞02]]; 6; 5; 2; 1; 1; 3; 3; 1+2]

Unlike the previous cases, the Cayley diagram contains loops, namely that one of the edges from a coset Nt_{i}t_{j} is to itself. This is what the 1 + 2 of the diagram is meant to indicate. The Cayley graph also contains multiple edges, for example ∞01 ∼ ∞04 ∼ 0∞ means that ∞0 has two joins to 0∞. (Of course, the regular graph contains no loops or multiple edges.) The regular graph is not bipartite either. The Coxeter diagram presentation corresponding to this symmetric presentation is given in Section 1.5.

4.1.4 A presentation for A_{7}

Here, we consider a progenitor with proper monomial action, namely 3^{⋆7}:_{m} L_{3}(2). Usually, such progenitors are much harder to deal with from a double coset enumeration point of view than their permutation counterparts. However, the coset enumeration we shall do here is relatively straightforward.

Now A_{7} is generated by elements t_{i} ∼ (1−i, 2−i, 4−i), where i may be taken to lie in 𝔽_{7}. These are monomially permuted by a subgroup ⟨(0, 1, 2, 3, 4, 5, 6), (2, 6)(4, 5)⟩ = ⟨(0, 6, 5, 4, 3, 2, 1), (0, 3)(5, 6)⟩ ≅ L_{3}(2). Thus we intend to take 3^{⋆7}:_{m} L_{3}(2) as our progenitor which has the presentation below:

⟨x, y, t | x^{7} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{3} = [t, y] = (t^{x}y)^{2} = (t^{x^{3}}y)^{2}⟩.

We realise x, y and t as shown in the table below.

Element Realisation in 3^{⋆7}:_{m} L_{3}(2) Realisation in A_{7}
x  (0, 1, 2, 3, 4, 5, 6)  (0, 6, 5, 4, 3, 2, 1)
y  (1, ¯1)(3, ¯3)(2, ¯6)(4, 5)  (0, 3)(5, 6)
t  t_{0}  (1, 2, 4)
t^{x^{i}}  t_{i}  (1−i, 2−i, 4−i)

We find, for instance, that (1, ¯1)(0, 3, ¯0, ¯3)(2, 6, 4, ¯5)t_{2}^{−1} [∼ (0, 3, 6)(2, 5, 4)] has order 3. Thus we consider the group:

G = \frac{3^{⋆7}:_{m} L_{3}(2)}{t_{2}t_{6}t_{4} = (1, ¯1)(0, ¯3, ¯0, 3)(2, ¯5, 4, 6)},

with a view to showing that it is A_{7}. (In terms of the above x, y and t, the extra relation is equivalent to ([x, y]t^{−x^{2}})^{3} = 1.)

Now N ≅ L_{3}(2) acts (sharply) transitively on the 168 = 14 × 12 ordered pairs (t, τ), where t and τ are symmetric generators such that ⟨t⟩ ≠ ⟨τ⟩. Thus all cosets (of N) of length at least 2 in the symmetric generators reduce to a coset of length at most 1 since we have t_{2}t_{6} ∼ t_{4}^{−1}. Therefore, the only cosets of N in G are ⋆ = N, i = Nt_{i} and ī = Nt_{i}^{−1}. However, we have already seen that A_{7} is an image of G and so |G : N| ⩾ 15. Thus it follows that the 15 cosets N, Nt_{i}, Nt_{i}^{−1} are distinct and that |G : N| = 15. This completes the proof that G ≅ A_{7}.

We realise G as permutations on 15 points (the cosets of N) as follows:

\begin{matrix}x ∼ & (0, 1, 2, 3, 4, 5, 6)(¯0, ¯1, ¯2, ¯3, ¯4, ¯5, ¯6), \\ y ∼ & (1, ¯1)(3, ¯3)(2, ¯6)(¯2, 6)(4, 5)(¯4, ¯5), \\ and t ∼ & (⋆, 0, ¯0)(1, 4, 2)(3, ¯5, ¯2)(5, ¯6, ¯1)(6, ¯3, ¯4). \end{matrix}

Here, we have used the abbreviations ¯3 for Nt_{3}^{−1} and so on.

The regular graph constructed on the right cosets of N in G with joins Nw—Ntw, where t is a symmetric generator, is just the complete graph on 15 points. Although G preserves this structure, it is not the full automorphism group of it, the full automorphism group being S_{15} in which G has index 518 918 400. We thus seek a different structure on the cosets of N in G that would give more insight into the action of G on 15 points. The geometry we define has 15 vertices (the Nw) and 35 = 15.7/3 ‘edges’ Nw—Niw—Nīw, each incident with 3 vertices. Note that this geometry is preserved by G acting by right multiplication on the cosets.

Now suppose there are two ‘edges’ that intersect in two points (vertices). Then, without loss, these edges are Nw—Nt_{i}w—Nt_{i}^{−1}w and Nv—Nt_{j}^{±1}v—Nt_{j}^{∓1}v where Nw = Nv and Nt_{i}w = Nt_{j}^{±1}v. Thus wv^{−1}, t_{i}wv^{−1}j^{∓1} ∈ N and so t_{j}^{±1} ∼ t_{i}wv^{−1} ∼ t_{i}^{wv^{−1}}. Since the cosets of length 1 are distinct we have that t_{i}^{wv^{−1}} = t_{j}^{±1} and so t_{i}^{−wv^{−1}} = t_{j}^{∓1} from which it follows that Nt_{i}^{−1}w = Nt_{j}^{±1}v, i.e., the edges intersect in three points and they are equal. Thus, using a pigeonhole principle, we see that these edges form a Steiner system S(2, 3, 15) on the 15 vertices. This Steiner system is familiar, as we shall see below.

Let us consider these triples a little further. First of all, we claim that they give rise to a solution of the Kirkman Schoolgirl Problem as shown below.

Day 0 ⋆0¯0 142 3¯5¯2 5¯6¯1 6¯3¯4
Day 1 ⋆1¯1 253 4¯6¯3 6¯0¯2 0¯4¯5
Day 2 ⋆2¯2 364 5¯0¯4 0¯1¯3 1¯5¯6
Day 3 ⋆3¯3 405 6¯1¯5 1¯2¯4 2¯6¯0
Day 4 ⋆4¯4 516 0¯2¯6 2¯3¯5 3¯0¯1
Day 5 ⋆5¯5 620 1¯3¯0 3¯4¯6 4¯1¯2
Day 6 ⋆6¯6 031 2¯4¯1 4¯5¯0 5¯2¯3

The Kirkman Schoolgirl Problem concerns a school with 15 (girl) pupils who go out for a walk each day in sets of 3. The object is to ensure that in the course of the week, each girl walks with each other girl. In our solution, we have labelled the girls with elements of {⋆, 0, …, 6, ¯0, …, ¯6} and the triples are triples of the Steiner system S(2, 3, 15) we have already constructed. Observe, however, that each of the triples can actually be written in the form {w, wi, wī} for some w ∈ {⋆, 0, …, 6, ¯0, …, ¯6} and some symmetric generator i. In our solution, on Day i, the triples of girls are {w, wi, wī}.

We now give a correspondence between between our 15 cosets and the (projective) points of the projective 3-space over 𝔽_{2} as shown below.

\begin{matrix}⋆ ∼ & ⟨(1, 0, 0, 0)⟩, & 1 ∼ & ⟨(0, 0, 1, 0)⟩, & ¯1 ∼ & ⟨(1, 0, 1, 0)⟩, & 4 ∼ & ⟨(0, 0, 1, 1)⟩, & ¯4 ∼ & ⟨(1, 0, 1, 1)⟩, \\ 0 ∼ & ⟨(0, 1, 0, 0)⟩, & 2 ∼ & ⟨(0, 0, 0, 1)⟩, & ¯2 ∼ & ⟨(1, 0, 0, 1)⟩, & 5 ∼ & ⟨(0, 1, 1, 1)⟩, & ¯5 ∼ & ⟨(1, 1, 1, 1)⟩, \\ ¯0 ∼ & ⟨(1, 1, 0, 0)⟩, & 3 ∼ & ⟨(0, 1, 1, 0)⟩, & ¯3 ∼ & ⟨(1, 1, 1, 0)⟩, & 6 ∼ & ⟨(0, 1, 0, 1)⟩, & ¯6 ∼ & ⟨(1, 1, 0, 1)⟩. \end{matrix}

Observe that the (35) triples of our Steiner system S(2, 3, 15) are (the 35) lines in the above projective 3-space over 𝔽_{2}. Thus, if we identify our cosets with the nonzero vectors of 𝔽_{2}^{4} (so ¯3 ∼ (1, 1, 1, 0), etc.), we see that the automorphisms of our S(2, 3, 15) are in 1–1 correspondence with the invertible linear transformations of 𝔽_{2}^{4}, so that this automorphism group is GL_{4}(2) ≅ L_{4}(2) of order 20 160 = 8 × |G|.

We embed our generators G ≅ A_{7} into L_{4}(2) (regarded as GL_{4}(2)) as follows:

x ∼ [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{matrix} ] , y ∼ [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \end{matrix} ] and t ∼ [ \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ] .

Note that since L_{4}(2) is simple and contains G to index 8 we must have an embedding L_{4}(2) ⩽ A_{8}. Thus we obtain the isomorphism L_{4}(2) ≅ A_{8} by equality of orders.

Note. Just as there is more than one way to skin a cat, there is more than one way to show the isomorphism between A_{8} and L_{4}(2). We sketch another way below. It is easily shown that:

⟨x_{i} : 1 ⩽ i ⩽ n | x_{i}^{3} = (x_{i}x_{j})^{2} = 1 for all i ≠ j⟩ ≅ A_{n+2},

where x_{i} ∼ (i, n+1, n+2). From this, it follows easily that

⟨x, y | x^{3} = y^{6} = (xy)^{7} = 1 = [x, y]^{2} = (xx^{y^{2}}) = [x, y^{3}]^{2}⟩ ≅ A_{8},

where x ∼ (1, 2, 3) and y ∼ (1, 2)(3, 4, 5, 6, 7, 8). (The relation [x, y^{3}]^{2} = 1 is, in fact, redundant.) However the matrices:

x ∼ [ \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{matrix} ] and y ∼ [ \begin{matrix}0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \end{matrix} ] ,

which are written over 𝔽_{2}, satisfy the above presentation for A_{8}. Since A_{8} is simple, it follows that we have the embedding A_{8} ⩽ GL_{4}(2) ≅ L_{4}(2). The isomorphism A_{8} ≅ L_{4}(2) then follows from the equality of the orders of the two groups.

4.2 The symmetric groups

Here, we give symmetric presentations of the symmetric groups. They were originally due to Curtis [13] and for n ⩾ 2 the presentations are as follows:

\frac{2^{⋆n}:S_{n}}{[(0, 1)t_{0}]^{3} = 1} ≅ S_{n+1}.

Of course, these presentations can be derived from the familiar Coxeter presentations of the symmetric groups, a topic that will recur in more detail throughout the thesis, see Section 7.5 for example.

4.3 The alternating groups

In this section we present two approaches to proving symmetric presentations.

4.3.1 The Moore presentation of A_{n+2}

We now prove a classical presentation of A_{n+2} given by E. H. Moore. Both proofs use induction. The first proof uses coset enumeration, is known anyway, and we do not give full details. The second proof is more interesting.

Theorem 4.3.1. For all n ⩾ 0 We have that:

G = ⟨t_{i} : 1 ⩽ i ⩽ n | t_{i}^{3} = 1 = (t_{i}t_{j})^{2} for all i < j⟩ ≅ A_{n+2},

where t_{i} ∼ (∞, 0, i) for all i.

The theorem obviously holds when n = 0 or 1. Thus we may suppose that n ⩾ 2 and assume the inductive hypothesis that H = ⟨t_{1}, t_{2}, …, t_{n−1}⟩ ≅ A_{n+1}. Note that the permutations given in the theorem obviously satisfy the presentation and generate A_{n+2}.

For the first proof, we simple enumerate the n+2 cosets of H in G. These are: H, Ht_{n}, Ht_{n}^{−1} and Ht_{n}t_{i}^{−1} for all i such that 1 ⩽ i ⩽ n−1. We leave it to the reader to verify that there no more cosets. (These cosets must then be distinct since we know an image Ḡ of G such that |Ḡ : H̄| ⩾ n+2.)

For the second proof we consider the group E obtained by adjoining an automorphism α to G such that α^{2} = 1 and for all i we have t_{i}^{α} = t_{i}^{−1} (or, equivalently, (t_{i}α)^{2} = 1). Then E has a homomorphism onto S_{n+2} with G mapping onto A_{n+2} as above and α ∼ (∞, 0). Since α is an automorphism, we have that G ≅ A_{n+2} if and only if E ≅ S_{n+2}. Now consider the group N = ⟨H, α⟩ ⩽ E. By the inductive hypothesis, we have N ≅ S_{n+1}. Let u_{∞} = u = αt_{n}. Then for i ∈ {1, …, n−1} we have t_{i}α.αt_{i} = t_{i}t_{n} which has order 2, and since αt_{n} and t_{i}α are involutions it follows that u commutes with t_{i}α. Now t_{i}α corresponds to (0, i) ∈ N and so u commutes with N^{∞} = Stab_{N}(∞) of index n+1 in N. In particular, u = u_{∞} has just n+1 images under N, and these may be labelled u_{i} for i ∈ {∞, 0, 1, …, n−1}. Also αu = α.αt_{n} = t_{n} has order 3, and we may write αu as (∞, 0)u_{∞}. Note also that E = ⟨N, u⟩. Therefore, E is a homomorphic image of:

\frac{(⟨u_{∞}⟩ ⋆ ⟨u_{0}⟩ ⋆ … ⋆ ⟨u_{n−1}⟩):N}{[(∞, 0)u_{∞}]^{3} = 1} ≅ \frac{2^{⋆(n+1)}:S_{n+1}}{[(∞, 0)u_{∞}]^{3} = 1},

a group that we have already shown to be S_{n+2}. This establishes the result.

4.3.2 Some related presentations

The Moore presentation above exhibits A_{n+2} as a group generated by n elements of order 3, called t_{i} say, such that ⟨t_{i}, t_{j}⟩ ≅ A_{4}. (Thus for all i < j, we have that t_{i}t_{j} or t_{i}t_{j}^{−1} has order 2.) It is interesting to ask which other groups can arise this way, in particular, those groups with presentations of the form:

G = ⟨t_{i} : 1 ⩽ i ⩽ n | t_{i}^{3} = 1 = (t_{i}t_{j}^{w_{ij}})^{2} for all i ≠ j⟩,

where w_{ij} = w_{ji} ∈ {1, −1} for all i ≠ j. For example, if we replace (t_{n_{1}}t_{n})^{2} = 1 in the above presentation for A_{n+2} by (t_{n_{1}}t_{n}^{−1})^{2} = 1 we obtain 2^{n}:A_{n+1}. Note that we are only interested in those cases that give rise to genuinely ‘new’ presentations and not those that can be obtained simply by permuting and/or inverting (some of) the generators.

We give some more intersting presentations below. In each case, generators with a subscript involving i should be regarded as being labelled with an element of 𝔽_{5}. We have:

⟨t_{i} : i ∈ 𝔽_{5} | t_{i}^{3} = 1 = (t_{i}t_{i+1})^{2} = (t_{i}t_{i+2}^{−1})^{2} for all i ∈ 𝔽_{5}⟩ ≅ U_{4}(2),

⟨t_{∞}, t_{i} : i ∈ 𝔽_{5} | t_{∞}^{3} = t_{i}^{3} = 1 = (t_{∞}t_{i})^{2} = (t_{i}t_{i+1})^{2} = (t_{i}t_{i+2}^{−1})^{2} for all i ∈ 𝔽_{5}⟩ ≅ S_{6}(2),

and

⟨t_{a}, t_{b}, t_{i} | t_{a}^{3} = t_{b}^{3} = t_{i}^{3} = 1 = (t_{a}t_{b})^{2} = (t_{a}t_{i})^{2} = (t_{b}t_{i})^{2} = (t_{i}t_{i+1})^{2} = (t_{i}t_{i+2}^{−1})^{2}⟩ ≅ 2·O_{8}^{+}(2).

4.3.3 An amalgam for A_{n+1}

For n ⩾ 3, the alternating group A_{n+1} is generated by three copies of A_{n} (the stabilisers of three distinct points in the natural n+1 point permutation representation of A_{n+1}) intersecting as shown in the diagram below. We may ask when this characterises the alternating groups A_{n+1}. The answer is: it does for all n ⩾ 5, but not when n = 3 or 4.

[figure: M_{1} ≅ A_{n}; M_{12} ≅ A_{n−1}; M_{2} ≅ A_{n}; M_{13} ≅ A_{n−1}; M_{23} ≅ A_{n−1}; M_{3} ≅ A_{n}; M_{123} ≅ A_{n−2}]

For n = 3 (the universal realisation of) this amalgam is just the free product 3⋆3⋆3 ≅ 3^{⋆3} and is therefore infinite. The cases when n ⩾ 5 are dealt with in the theorem below and the case n = 4 seems to be a bit tricky, and we deal with this by way of a presentation in the next paragraph.

In the case n = 4, we may take M_{ij} = ⟨t_{k}⟩ ≅ 3, where {i, j, k} = {1, 2, 3}. Thus M_{i} = ⟨t_{j}, t_{k}⟩ ≅ A_{4} which forces t_{i}t_{j} or t_{i}t_{j}^{−1} to have order 2. Inverting the t_{i} if necessary, we may assume that t_{1}t_{2} and t_{1}t_{3} have order 2. Let G be the universal realisation of the amalgam. Then G is generated by t_{1}, t_{2} and t_{3} and satisfies one of the two presentations below:

G ≅ ⟨t_{1}, t_{2}, t_{3} | t_{1}^{3} = t_{2}^{3} = t_{3}^{3} = 1 = (t_{1}t_{2})^{2} = (t_{1}t_{3})^{2} = (t_{2}t_{3}^{±1})^{2}⟩.

The option with (t_{2}t_{3})^{2} = 1 gives rise to A_{5} and the option with (t_{2}t_{3}^{−1})^{2} = 1 gives rise to 2^{1+2}:A_{4} ≅ E_{8}:A_{4}. Thus there are two amalgams with the above diagram when n = 4, and they both give rise to finite universal realisations.

Theorem 4.3.2. For n ⩾ 5, the universal realisation of the above amalgam is A_{n+1}.

Proof. Let G be the universal realisation of the amalgam. Let Γ be the (coset) graph with vertices M_{i}g such that two distinct cosets are joined just when their intersection is non-empty. Thus Γ is tripartite since M_{i}g can only be joined to M_{j}h when i ≠ j.

Note that in the general case, if i ≠ j then M_{i} is joined to just |M_{i} : M_{ij}| cosets M_{j}g and that these take the form M_{j}τ where τ is an element of a right transversal of M_{ij} in M_{i}. Thus, in general M_{i}g and M_{j}h are joined just when M_{i} ∩M_{j}hg^{−1} = (M_{i}g ∩M_{j}h)g^{−1} ≠ ∅, and we may take hg^{−1} = τ to be a transversal element as in the previous sentence. So if i ≠ j, M_{i}g is joined to just |M_{i} : M_{ij}| cosets M_{j}h.

Also, we note that Γ is connected. To see this, we consider the coset M_{i}g where g = g_{1}g_{2} … g_{m} where g_{1} ∉ M_{i} and g_{ℓ} ∈ M_{a_{ℓ}} such that a_{ℓ} ∈ {1, 2, 3} and a_{ℓ} ≠ a_{ℓ−1} for all appropriate ℓ. Then

M_{i}g = M_{i}g_{1} … g_{m}—M_{a_{1}}g_{2} … g_{m}— ⋯ —M_{a_{m−1}}g_{m}—M_{a_{m}},

is path in Γ connecting M_{i}g to one of the M_{j}. However, the M_{j} are pairwise joined, so Γ is connected.

We now show that there are at most n cosets M_{1}g at distance 2 from M_{1} (in Γ) and that they form a single orbit under right multiplication by M_{1}. Note that a path of length 2 connecting M_{1} and M_{1}g takes the form M_{1}—M_{j}h—M_{1}τh where j ∈ {2, 3}, h ∈ M_{1} and τ ∈ M_{j} \ M_{1}. Thus it is enough to show that the M_{1}τ for τ ∈ (M_{2} ∪ M_{3}) \ M_{1} form a single orbit under M_{1} and that, for some such τ, the index in M_{1} of Stab_{M_{1}}(M_{1}τ) is n.

Let j ∈ {2, 3}. Now for h ∈ M_{1j} and τ_{1}, τ_{2} ∈ M_{j}, M_{1}τ_{1} = M_{1}τ_{2}h just when M_{1} = M_{1}τ_{2}hτ_{1}^{−1} which is just when M_{1j}τ_{1} = M_{1j}τ_{2}h. Therefore, since n ⩾ 4, M_{j} acts 2transitively on the cosets of M_{1j}, and so whenever τ_{1}, τ_{2} ∈ M_{j} \ M_{1}, we can always find h ∈ M_{1j} such that M_{1}τ_{1} = M_{1}τ_{2}h. Thus for j ∈ {2, 3}, M_{1j}, and hence M_{1}, has just one orbit on the M_{1}τ such that τ ∈ M_{j} \ M_{1}. But there are cosets M_{1}τ such that τ ∈ M_{23} \ M_{1} = (M_{2} \ M_{1}) ∩ (M_{3} \ M_{1}) and so M_{1} acts transitively on the cosets M_{1}τ such that τ ∈ (M_{2} ∪ M_{3}) \ M_{1}.

Let τ ∈ M_{23} \ M_{1}. Then Γ has paths M_{1}—M_{2}—M_{1}τ and M_{1}—M_{3}—M_{1}τ which we label as P_{1} and P_{2} respectively. Now S_{1} = Stab_{M_{1}}(P_{1}) = M_{12} ∩ M_{1}^{τ} = M_{12} ∩ M_{12}^{τ}, which implies that Stab_{M_{1}}(P_{1}) has index n−1 in M_{12}. Similarly, S_{2} = Stab_{M_{1}}(P_{2}) has index n−1 in M_{13}. Now S_{1} = S_{2} only if S_{1} = S_{2} = M_{123}, which implies that M_{1}τh = M_{1}τ for all h ∈ M_{123}, that is τhτ^{−1} ∈ M_{123} for all h ∈ M_{123}, a situation which can only arise if n ⩽ 4. (If n ⩾ 5 then M_{123} ≅ A_{n−2} is self-normalising in M_{23} ≅ A_{n−1}.) Now S_{1} is conjugate in M_{12} to M_{123} since both S_{1} and M_{123} are 2-point stabilisers in the action of M_{2} on the cosets of M_{12}. Similarly, S_{2} and M_{123} are conjugate in M_{13} and hence S_{1}, S_{2} and M_{123} are conjugate in M_{1}. Therefore, S_{1} and S_{2} are unequal 2-point (pointwise) stabilisers in M_{1} and so ⟨S_{1}, S_{2}⟩ ≅ A_{n−1} or A_{n}. But Stab_{M_{1}}(M_{1}τ) ⩾ ⟨S_{1}, S_{2}⟩, a group we have now shown has index at most n in M_{1}. This completes the proof that there are at most n cosets M_{1}g at distance 2 from M_{1} and that they fall into a single orbit under M_{1}.

Next we consider the unique coset C = M_{1}τ that is distance 2 from M_{1}, joined to M_{3} but not to M_{2}. Then C is fixed by M_{12}, which fixes the edge M_{1}—M_{2}, since C is uniquely determined by the condition that it is of type M_{1}g, of distance 2 from M_{1}, and not joined to M_{2}. But M_{23} fixes the edge M_{2}—M_{3}, and since C uniquely specified by the condition that is the type M_{1}g coset that is adjacent to M_{3} but not to M_{2} it follows that C is fixed by M_{23}. Thus C is fixed by the whole of M_{2} = ⟨M_{12}, M_{23}⟩. However, we know that the stabiliser of C = M_{1}τ is M_{1}^{τ}, and by equality of orders we obtain that M_{2} = M_{1}^{τ}. Similarly, the unique coset M_{1}h of distance 2 from M_{1} and M_{3} is fixed by M_{3} and we have M_{3} = M_{1}^{h}.

Thus we have shown that the n+1 cosets M_{1} and M_{1}g of distance at most 2 from M_{1} are closed under right multiplication by M_{1}, M_{2} and M_{3} and hence by G. This shows that M_{1} has index (at most) n+1 in G. This completes the determination of the order of G. To see that G acts as A_{n+1} on these cosets of M_{1}, we shall produce a few 3-cycles. Let M_{2} act on {1, …, n} with M_{12} = Stab_{M_{2}}(1). Then if we label the cosets as C_{1} = M_{1}, C_{n+1} = C, C_{n} = M_{1}(1, n, 2) and C_{j} = M_{1}(1, j, j+1) for 2 ⩽ j ⩽ n−1, then we see that (p, q, r) of M_{2} acts as the 3-cycle (C_{p}, C_{q}, C_{r}) on the C_{i} whichever (distinct) p, q, r ∈ {1…. , n} are chosen. □

4.3.4 An application

We now show how we may use the above amalgam to prove an infinite series of symmetric presentations for alternating groups. For n ⩾ 1, the alternating group A_{n+4} acting on {a, b, c, d, 0, 1, …, n−1} is generated by the elements:

t_{i} ∼ (a, b, c, d, i) and u_{i} ∼ (b, a, d, c, i),

where i ∈ Ω_{n} = {0, 1, …, n−1}. The symmetries of the set T = {t_{i}^{j}, u_{i}^{j}} in S_{n+4} form a group S_{n} × D_{8}, which is generated by a subgroup Sym(Ω_{n}) ≅ S_{n} which simply permutes the t_{i} and the u_{i} together with i ∼ (a, b, d, c), which squares the t_{i} and cubes the u_{i}, and σ ∼ (a, b)(c, d) which interchanges t_{i} and u_{i}. The intersection of this group of symmetries with A_{n+4} is ⟨Alt(Ω_{n}), (0, 1)i, σ, i^{2}⟩ ≅ (A_{n} × 2^{2}):2. (For n ⩾ 2, the generator i^{2} = [(0, 1)i]^{2} is redundant.) The D_{8} in the above S_{n} × D_{8} is generated by i and σ where i may be regarded as \sqrt{−1} ∈ ℂ and σ may be regarded as complex conjugation.

For n ⩾ 4 we have the symmetric presentation:

\frac{5^{⋆(n+n)}:_{m} (A_{n} × 2^{2}).2}{(σt_{0})^{3} = ((0, 1)iσt_{0}^{2})^{3} = 1} ≅ A_{n+4}.

For n = 3, this presentation gives the group 3·A_{7}. To prove this presentation, we use Magma to prove the case n = 4. We then use induction and the amalgam of Theorem 4.3.2 to deal with the general case. If we relax this presentation by removing the relation (σt_{0})^{3} = 1, we obtain the image A_{n+4} × S_{5} whenever n ⩾ 4. As a special case, for n = 8 we obtain the symmetric presentation:

\frac{5^{⋆(8+8)}:_{m} (A_{8} × 2^{2}).2}{(σt_{0})^{3} = ((0, 1)iσt_{0}^{2})^{3} = 1} ≅ A_{12}.

We shall see this again in Section 9.3.2.

Connected to this is the homomorphism

5^{⋆n}:_{m} (A_{n} × 2)·2 ⟶^{onto} A_{n+4}.

We may turn this into a presentation for n ⩾ 3 by requiring that subsets of 3 symmetric generators (from distinct T_{i}) generate A_{7} (and that ⟨t_{i}, t_{j}⟩ ≅ A_{6}) and that we append enough relations to ensure that (0, 1)σ ∈ ⟨T ⟩. Adding ((0, 1)σt_{0})^{5} = 1 ensures both that (0, 1)σ ∈ ⟨T ⟩ and that ⟨t_{i}, t_{j}⟩ ≅ A_{6}. With this relation, and for n ⩾ 3, we can ensure that ⟨t_{i}, t_{j}, t_{k}⟩ ≅ A_{7} by adding (0, 1, 2) = t_{0}^{−1}t_{1}t_{2}^{−1}t_{0}. This presentation is proved in the same manner.

Computer packages

5.1 Introduction

In obtaining the results presented in this thesis, we have made extensive use of a computer, in particular various computer algebra packages. In the main, we have used Cayley V3.8.4 [6, 7] and Magma [2]. (The present version we have is Magma V1.20–1.) We used Cayley to obtain most of the tabulated results in Chapter 7 which also appeared in my M. Phil. [3], but for the rest of this thesis (the parts we did more recently—including the rest of Chapter 7) we have used Magma. We have also ‘dabbled’ in GAP [36] and the MeatAxe^{1} but we have not made any serious use of either. Our main use of GAP has been for character table calculations since GAP has an extensive library of stored character tables.

In this chapter, we have concentrated on group isomorphism-type identification techniques, giving Magma code where appropriate. Computer code (almost exclusively Magma) is given in the typewriter font. All the described packages except Cayley are case sensitive and in all the packages (except the MeatAxe) most lines of code must be terminated by a semi-colon.

We refer the reader to the Magma handbook [2] for details of variants of and optional parameters to the commands exhibited here. Deciding which of the optional parameters to use and what to set them to is something of an art.

5.2 Finitely presented groups

We consider the group G with presentation as shown below:

\begin{matrix}G = ⟨x, y, t | x^{7} = & y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{3} \\ = & [xy, t^{x^{4}}] = [y^{x^{4}}, t] = (x^{3}yt)^{7}, y = [x^{2}, t^{x}]^{2}⟩. \end{matrix}

1There are many versions of the MeatAxe in existence. In this context MeatAxe refers to any of them. If the context means that we are referring to a specific MeatAxe, then this is not Parker’s original Fortran-MeatAxe but M. Ringe’s translation of this into the C-MeatAxe. The expression “Integral-MeatAxe” refers to Parker’s new integral MeatAxe [32]. There are also various MeatAxe facilities in Magma.

In terms of symmetric presentations, the relations up to (and including) 1 = [y^{x^{4}}, t] define the progenitor

3^{⋆7}:L_{3}(2),

where x ∼ (0, 1, 2, 3, 4, 5, 6), y ∼ (2, 6)(4, 5) and t ∼ t_{0}. In fact, we have:

G = \frac{3^{⋆7}:L_{3}(2)}{(2, 6)(4, 5) = [t_{1}t_{3}^{−1}]^{2}, [(0, 3)(2, 6, 4, 5)t_{2}]^{7} = 1} ≅ 2·J_{2},

a result of Curtis [18]. Two important subgroups in which we will be interested are the control subgroup, N = ⟨x, y⟩ ≅ L_{3}(2), and H = ⟨t_{0}, t_{1}, t_{2}, t_{5}⟩ ≅ U_{3}(3). (Note that we do not yet know that G ≅ 2·J_{2} or that H ≅ U_{3}(3)—these facts will be shown in due course.) We specify the finitely presented group G as above on Magma by typing:

   G<x,y,t>:=Group<x,y,t|x^7,y^2,(x*y)^3,(x,y)^4,t^3,(x*y,t^(x^4)),
   (y^(x^4),t),(x^3*y*t)^7,y=(x^2,t^x)^2>;

where G, x, y and t correspond to G, x, y and t respectively.

5.2.1 Other groups used in this chapter

Occasionally, to illustrate a particular method, we shall have recourse to consider a group other than 2·J_{2}. The groups actually used are given below. Throughout this chapter, G, N, H, x, y and t will refer to subgroups and elements of 2·J_{2} unless otherwise stated.

We consider the group G with presentation as shown below:

G = ⟨x, y, t | x^{8} = y^{4} = (xy)^{3} = (x^{−3}y)^{2} = (x^{4}y^{2})^{5} = 1 = t^{2} = [t, xy^{−1}] = [t, x^{2}y] = (tx)^{5}⟩.

In terms of symmetric presentations, this is the group:

G = \frac{2^{⋆10}:M_{10}}{[(1, 2, 3, 6, 5, 4, 7, X)(8, 9)t_{1}]^{5} = 1} ≅ L_{3}(4):2_{1},

where x ∼ (1, 2, 3, 6, 5, 4, 7, X)(8, 9), y ∼ (1, 2, X, 3)(6, 9, 8, 7) and t ∼ t_{1}. This is a result of Curtis [18] and is seen again in Chapter 8. Two important subgroups in which we will be interested are the control subgroup, N = ⟨x, y⟩ ≅ M_{10}, and H = ⟨t_{1}, t_{2}, t_{3}, t_{9}⟩ ≅ L_{3}(2):2. (Note that we do not yet know that G ≅ L_{3}(4):2_{1} or that H ≅ L_{3}(2):2—these facts will be shown in due course.)

We shall also consider the group G with presentation as shown below:

G = ⟨x, y, t | x^{8} = y^{4} = (xy)^{3} = (x^{−3}y)^{2} = (x^{4}y^{2})^{5} = 1 = t^{3} = [t, xy^{−1}] = t^{x^{2}y}t = (tx)^{5}⟩,

which, in symmetric presentation terms, becomes:

G = \frac{3^{⋆10}:_{m} M_{10}}{[(1, 2, 3, 6, 5, 4, 7, X)(8, 9)t_{1}]^{5} = 1} ≅ 3_{2}·U_{4}(3),

where x ∼ (1, 2, 3, 6, 5, 4, 7, X)(8, 9), y ∼ (1, 2, X, 3)(6, 9, 8, 7) and t ∼ t_{1}. We shall see this presentation again in Chapter 8. Two important subgroups in which we will be interested are the control subgroup, N = ⟨x, y⟩ ≅ M_{10}, and H = ⟨t_{1}, t_{2}, t_{3}, t_{9}⟩ ≅ U_{3}(3). (Note that we do not yet know that G ≅ 3_{2}·U_{4}(3) or that H ≅ U_{3}(3)—these facts will be shown in due course.)

5.3 Subgroups and order of a group

Three subgroups of G in which we are interested are the control subgroup N = ⟨x, y⟩ ≅ L_{3}(2), K = ⟨N, t_{0}^{−1}t_{1}⟩ which is a larger subgroup than N and may or may not be the whole of G, and H = ⟨t_{0}, t_{1}, t_{2}, t_{5}⟩ which is the subgroup of G generated by an oval of symmetric generators. We specify this in Magma as follows:

   N:=sub<G|x,y>;
   K:=sub<G|N,(t,x)>;
   H:=sub<G|t,t^x,t^(x^2),t^(x^5)>;

The first step in trying to identify a group is to establish its order. The naïve way to do this would be to type print Order(G). However, this would be a trifle foolish since it would require that we enumerate the 1 209 600 cosets of the trivial subgroup.^{2} Instead, we may proceed more expediently if we enumerate the cosets in G of a subgroup, L say, whose order is known (and preferably as large as possible) and use Lagrange’s Theorem that |G| = |G : L|.|L| to deduce the order G. We may accomplish this by using

   print Index(G,N);

which easily gives the answer that |G : N| = 7 200. But we arranged that there were sufficient relations between x and y to ensure that N = ⟨x, y⟩ was (a homomorphic image of) L_{3}(2) of order 168. Thus G has order (at most) 7 200 × 168 = 1 209 600 = 2.|J_{2}|.

5.4 Obtaining a permutation representation

To deal with the group, a permutation representation would be useful. A permutation representation of G over the right-cosets of N, of degree 7 200, would be on the large side for such a (relatively) small group. Therefore, we may look at some other easily identifiable groups. It is often useful to consider a subgroup generated by a subset of the symmetric generators. Indeed, this is what we do here, for if H = ⟨t_{0}, t_{1}, t_{2}, t_{5}⟩, it can be shown (quite easily) that H is a homomorphic image of:

\frac{3^{⋆4}:S_{4}}{(2, 3) = [t_{0}t_{1}^{−1}]^{2}, [(0, 1, 2, 3)t_{0}]^{7} = 1} ≅ U_{3}(3).

If we take the permutation action of G over H we obtain a permutation group of degree 200 which turns out to be faithful. We obtain permutation representations on Magma as follows:

   f,gp<x1,y1,t1>:=CosetAction(G,H);

2The figure 1 209 600 is somewhat retrospective—at this stage one would have no idea what the group order was.

In the above, gp denotes the image of G as a permutation group on the right-cosets of H, f denotes the homomorphism from G to gp, and x1, y1 and y1 are the images of x, y and t under f. The commands:

   [print ]Degree(gp);
   [print ]Order(gp);

find the degree and order of the permutation group gp. There are many commands for permutation groups not described here. In our case, we find that gp has degree 200 and order 1 209 600, hence showing that the permutation representation is faithful, and that |G| = 1 209 600.

Another example of a useful subgroup to try to get a permutation representation over is K = ⟨N, t_{0}^{−1}t_{1}⟩. In this case, K is another copy of U_{3}(3) and so G has a faithful permutation representation over K also. However, the fortuitous set of circumstances where a subgroup of low index is easily identified may not arise. In such a case, one has to be constructed. In the first instance, the function:

   [ss:=]LowIndexSubgroups(G,<m,m>:Subgroup:=J,Limit:=ℓ);

where m is the index of a subgroup over which we expect to obtain a faithful permutation representation might be of some use. This finds up to ℓ conjugacy class representatives of subgroups of index m in G containing a conjugate of J. (If Subgroup:=J is omitted then J = 1 is assumed.) It is usually best to set ℓ = 1. We note that there are occasions when one might wish to know every subgroup of very low index (say ⩽ 10) but we shall say more about this when relevant. The utility of this function is highly presentation dependent, but it is usually restricted to index at most 30.

If the above has failed to yield a low index subgroup, one has to try a ‘random’ selection of likely-looking subgroups.

5.5 Cutting a group down to size

Having obtained a permutation representation of low degree for G (or for a homomorphic image of G obtained by quotienting out by a subgroup of small order), we are in a position to perform operations to reveal the structure of the group. From now on we shall use G to denote the (permutation group) image of the (original) finitely presented group. Of obvious utility are the functions:

   IsSimple(G);
   CompositionFactors(G);
   Centre(G);
   DerivedSubgroup(G);
   DerivedSeries(G);

along with Degree(G) and Order(G) which are fairly self-explanatory and may be used for degrees up to several thousand. It is alleged that CompositionFactors(G) arranges the composition factors of G in the order in which they appear in a composition series of G, but the function seems fallible in this respect.

Two more intrusive (and useful) functions are:

   Classes(G);
   NormalLattice(G);

but these are usually limited to degrees of a few hundred.

From a knowledge of all the normal subgroups a group possesses, or equivalently, all the chief series a group has, it is possible to deduce the structure of a group for a large class of groups including many of those we are likely to obtain as images of our progenitors. (Such theorems as those for minimal normal subgroups come into their own here—namely, that a minimal normal subgroup of a group is a direct product of isomorphic simple groups. In particular, if a minimal normal subgroup is a p-group then it must be elementary abelian.)

5.5.1 Identifying the ‘rough’ shape of our groups

If the permutation group G is isomorphic to one of the groups we considered in Section 5.2 then CompositionFactors(G) gives the possibilities G ≅ p×S, p·S (a perfect group) or S.p (with trivial centre) where S is non-abelian simple and p ≅ C_{p} is cyclic of prime

order. The commands Order(Centre(G)) and Index(G,DerivedSubgroup(G)) then

distinguish these three possible cases. Here, the minimal possible degrees for G are 200, 42 and 378 respectively.

We thus obtain the rough shapes 2·J_{2}, L_{3}(4).2 and 3·U_{4}(3) respectively. The first case gives rise to a unique isomorphism type, but the other two do not. In the second case, the outer 2 acts as a non-trivial outer automorphism on the L_{3}(4), this gives rise to the possibilities L_{3}(4):2_{1}, L_{3}(4):2_{2} and L_{3}(4):2_{3} in the second case. In the third case, we have the possibilities 3_{1}·U_{4}(3) and 3_{2}·U_{4}(3). (See the Atlas for the meanings of L_{3}(4):2_{1}, 3_{1}·U_{4}(3) and so on.)

For the second case, if we only knew that G had normal subgroups of orders 1, 20 160 and 40 320, then we would also have to consider the case G ≅ S_{8} ≅ A_{8}:2.

5.5.2 How to identify automorphisms and/or covers

Our favourite method of identifying which outer automorphism we have in a group S.d is to centralise it in the simple group S. The centralisers in S of outer involutions in groups L_{3}(4).2 are U_{3}(2) ≅ M_{9} ≅ 3^{2}:Q_{8}, L_{3}(2) and A_{5} in the cases when these outer involutions are 2_{1} = 2_{f×g}, 2_{2} = 2_{f} and 2_{3} = 2_{g} respectively. (The corresponding centralisers when S ≅ A_{8} are S_{6} and S_{4} × 2.) Thus we may distinguish the various groups L_{3}(4).2 and A_{8}.2 by the order of the centraliser of an outer involution.

It is not so easy to identify whether a group is A_{1}·K or A_{2}·K for some group K and abelian group A (with A_{1} ≅ A_{2} ≅ A) where A_{i} ⩽ (A_{i}·K)′ for i = 1 and 2. However, when K is simple, the situation does not arise very often. The possible cases are (A, K) ≅ (4, L_{3}(4)), (12, L_{3}(4)), (3, U_{4}(3)), (6, U_{4}(3)), (12, U_{4}(3)) and (2, O_{4n}^{+}(q)) where n ⩾ 3 and q is odd. In these cases there are two isomorphism types of A·K. At present, we distinguish the different types of 3·U_{4}(3) by taking a permutation action of degree 378 on the cosets of U_{4}(2) and calculating

   #Orbits(Stabiliser(permutation group,1));

which is 7 and 5 in the cases 3_{1}·U_{4}(3) and 3_{2}·U_{4}(3) respectively.

The way we originally showed that the third group was 3_{2}·U_{4}(3) was to try to add an automorphism of type 2_{3}. (There is no such automorphism of 3_{1}·U_{4}(3).) In fact, such an automorphism turns out to be easy to find—we simply fix x and y and invert t.

5.5.3 Some general remarks

If a group G is primitive, which we may check using IsPrimitive(G), then its normal subgroups are transitive and the point stabiliser is maximal. Thus in a group G ≅ p^{m}:N, if the action of G on the cosets of N is primitive and faithful, we obtain that N acts irreducibly (but not necessarily absolutely irreducibly) on the p^{m}.

In a group with chief series 1 = G_{0} ◁ G_{1} ◁ G_{2} ◁ … ◁ G_{l} = G in which |G_{1}| = p and |G_{2}| = p^{m} where m ⩾ 3 (and p is prime) we have that G_{2} is elementary abelian or extraspecial. Extraspecial groups (for the order p^{m}, m odd) come in two isomorphism classes and may be distinguished by counting elements of orders 1, p and p^{2}. If p is odd then G_{2} must have exponent p (so G_{2} ≇ p\begin{matrix}1+(m−1) \\ − \end{matrix}).

5.5.4 What happens if a faithful (permutation) representation is hard to come by?

Consider the group:

G = \frac{2^{⋆3}:S_{3}}{[(0, 1, 2)t_{0}]^{15} = [(0, 1)t_{0}]^{8} = [(0, 1)t_{0}t_{2}]^{10} = 1} ≅ 2^{2}·L_{3}(4):S_{3}.

The order of G be be verified by coset enumeration of the cosets of H = ⟨(0, 2), (1, 2)t_{0}⟩ ≅ D_{30} followed by a permutation action of G on the cosets of H which has degree 16 128. Such a permutation representation of G precludes detailed structural analysis.

   Using LowIndexSubgroups(G,<42,42>:Limit:=1) quickly yields a subgroup J of in-

dex 42 in G such that the permutation action of G on the cosets of H gives the image L_{3}(4):S_{3} ≅ L_{3}(4):3:2_{3}. Knowing the order of G, we deduce that K, the kernel of this action has order 4. We then used LowIndexSubgroups(G,24) which almost instantaneously told us that there were just four classes of subgroups of index at most 24—one each of indexes 1, 2, 3 and 6. This was enough to show that G′′ was perfect of index 6 and hence G′′ ≅ 4·L_{3}(4) or 2^{2}·L_{3}(4). However, 4·L_{3}(4) has no outer automorphism of order 3 and so G′′ ≅ 2^{2}·L_{3}(4) and G ≅ 2^{2}·L_{3}(4):S_{3}.

In this case, the minimal faithful permutation representation of G has degree 336 over the cosets of PGL_{2}(9) × 2. We found that L = ⟨t_{0}, t_{1}t_{2}, (1, 2)⟩ ≅ PGL_{2}(9) × 2 and in the permutation action of G on the cosets of L it was (computationally) easy to verify all the above assertions.

In many similar examples, it is not necessary to go to such lengths, for it is often the case that for all images G of a given progenitor either G is perfect or G′ is perfect. This applies, for example, to all progenitors featured in Chapter 7 that have insoluble control subgroup except 3^{⋆5}:_{m} S_{5} where G′′ is perfect but G′ need not be. In the above case we were able to show that G′′ is ⟨T ⟩ and had the following “presentation”:

⟨t_{0}, t_{1}, t_{2} | t_{0}^{2} = t_{1}^{2} = t_{2}^{2} = 1 = (t_{0}t_{1}t_{2})^{5} = (t_{0}t_{1}t_{0}t_{2})^{5} = (t_{1}t_{2}t_{1}t_{0})^{5} … other relations⟩,

which was already enough to show that G′′ was perfect.

In the progenitor 2^{⋆4}:A_{4}, we came across an image G that we knew (or suspected) had shape L_{2}(8)^{2}:3 and was a subdirect product of (L_{2}(8):3)^{2}. We therefore constructed both subdirect products of (L_{2}(8):3)^{2} which were of shape L_{2}(8)^{2}:3 as intransitive 18point permutation groups and looked inside them for generators (not merely elements) satisfying our presentation. Thus we were able to determine that the 3 acted differently on the two L_{2}(8)’s. In fact, we can construct the 18-point representation of G by taking the sum (or product?!) of the two 9-point representations obtained from LowIndexSubgroups.

Actually, what we described in the previous paragraph is the original method we used to identify this group. At the moment, our preferred method for identifying this group (and others like it) is to take a Hall {7, 3}-group of the normaliser of a Sylow 7-subgroup. We find that it is 7^{1_{2}⊕1_{3}}:3, showing that G is the variety of L_{2}(8)^{2}:3 in which the 3 acts differently on the two L_{2}(8)’s. (In the case of L_{2}(8)^{2}:3 where the 3 acts the same on both L_{2}(8)’s, the required {7, 3}-group has structure 7^{1_{2}⊕1_{2}}:3.)

For the meanings of 7^{1_{2}⊕1_{3}}:3 and 7^{1_{2}⊕1_{2}}:3 the reader is referred to Appendix A. Here, all the reader needs to know is that both these groups have trivial centre, that 7^{1_{2}⊕1_{3}}:3 has five normal subgroups (of orders 1, 7, 7, 49 and 147) and that 7^{1_{2}⊕1_{2}}:3 has 11 normal subgroups (of orders 1, 7, 7, 7, 7, 7, 7, 7, 7, 49 and 147).

5.6 Some other structures

There facilities in Magma for computing with structures such as fields, matrix groups and permutation groups. We briefly describe these below.

5.6.1 Fields

There are standard functions on Magma for creating the ‘standard’ fields 𝔽_{q}, ℚ, ℚ(\sqrt{d}) and ℚ(e^{2πi/n}). We may also compute with an arbitrary number field, for example ℚ(^{3} \sqrt{2}), but not so efficiently as one of the above fields. For any of the fields (F) already mentioned, we may create the (very useful) fields F (y_{1}, y_{2}, …, y_{m}). The primitive roots of the more often used finite fields are computed using Conway polynomials, thus ensuring consistency between Magma and other computer algebra packages such as GAP and the MeatAxe.

(A Conway polynomial is a particular method for specifying a primitive element of 𝔽_{q} up to field automorphism.)

5.6.2 Groups

We can compute in Magma with finitely presented, matrix and permutation groups. There are in-built functions for creating ‘standard’ groups (for example the alternating, symmetric, linear, unitary and symplectic groups) in their ‘standard’ permutation and matrix representations. We have written Magma programs for performing basic operations with monomial groups over a wide range of rings, and also for creating these [monomial groups] from larger degree permutation groups.

As for the more ‘exotic’ representations of finite groups (including the sporadic simple groups), Magma has a small collection of these including Sims’s list of primitive groups of degree at most 50. However, this collection is not very extensive and a more comprehensive collection is provided by R. A. Wilson in his Atlas of Group Representations [40] which is available on the World Wide Web.

5.7 The computers used

Most research for this thesis was done using a Sun SPARCstation 20 (local name fourier) and a Sun SPARCstation ELC (local name sylow). We will sometimes give times taken for computations performed. These times are very approximate and it must be remembered that the time taken for a calculation will depend on which computer is used and on the load on the computer from other processes running.

We used fourier for any ‘heavy’ computations in this thesis, for example the coset enumerations and matrix constructions for HN in Chapter 9.

5.8 Note

During the period between the writing of this chapter and submission of the thesis, Magma was upgraded to V2.20–3. However, since we performed most of our calculations using V1.20–1 and earlier versions we have decided not to rewrite this thesis to conform to V2.20–3. In any case, the commands mentioned here still work in V2.20–3 and most (but not necessarily all) of the other information, such as approximate running times, is about the same. The only significant difference we have noticed between V2.20–3 and earlier versions of relevance to this thesis is that matrix group calculations over fields of characteristic zero (such as ℚ) are much faster. (There are also ‘cosmetic’ differences such as more commands and the fact that print is now unnecessary.)

Matrix representations

6.1 Introduction

Part of this thesis concerns itself with representations of various groups, especially the progenitors we are studying, but also some of their finite images. In particular, in Chapter 9, we shall construct a 133-dimensional representation of HN over ℚ(b5) where b5 = \frac{1}{2}(−1 + \sqrt{5}) has minimal polynomial z^{2} + z − 1. On the way to this construction we also construct a 56-dimensional representation of 2·HS:2 over ℚ(b5) and the 77-dimensional representation of HS:2 over ℚ. In this chapter, we explain some of the techniques we have employed to construct matrix representations of groups.

6.2 The basic technique

The technique we intend to employ is a variant of the amalgam technique described by Parker and Wilson [33] and pioneered by Norton and Parker [31] (in about 1980) when constructing the 112-dimensional representation of J_{4} over 𝔽_{2}. We describe this technique first. Here, we shall let G be the group whose representation we want to construct. Suppose that G is generated by (proper) subgroups H and K intersecting in L. Then G is a homomorphic image of a free amalgamated product H ⋆_{L} K. The situation is shown diagrammatically below.

[figure: G; H; K; L]

Fixing our notation throughout this section, we shall let F be a field, V = F^{n} and ρ : G → GL(V)(≅ GL_{n}(F)) be the representation we wish to construct. Without loss of generality, we may suppose that the representations we are considering that are irreducible over F are absolutely irreducible over F. We proceed roughly as follows:

(i) Determine the restrictions of ρ to H and K and construct representations f_{H} : H → GL(V) and f_{K} : K → GL(V) such that f_{H} is equivalent to ρ|_{H} and f_{K} is equivalent to ρ|_{K}.

(ii) Find generators for f_{H}(L) in f_{H}(H) and f_{K}(L) in f_{K}(K).
(iii) Change f_{K} to an equivalent representation f′_{K} so that f_{H}(L) = f′_{K}(L).

(iv) Change f′_{K} to an equivalent representation f′′_{K} while keeping f_{H}(L) = f′_{K}(L) = f′′_{K}(L) until we have ⟨f_{H}(H), f′′_{K}(K)⟩ ≅ G.

We describe the various steps in more detail below.

6.2.1 Choosing H and K

We should choose H and K in such a way that the restriction of ρ to L = H ∩ K has as few constituents as possible. This is so the number of degrees of freedom when doing the conjugation in Step (iv) above is as small as possible. We should also be able to construct the restrictions of ρ to H and K. We also prefer the restriction of ρ to each of H, K and L to break up as a direct sum of irreducible constituents—this is automatic for characteristic 0 representations of finite groups. We note that the requirements of this paragraph may conflict and so compromises may have to be made.

6.2.2 Constructing f_{H}(H) and f_{K}(K)

The general idea is that this is meant to be easy. However, making f_{H}(H) and f_{K}(K) can be fairly complex constructions in their own right. A general method for doing this is to construct a permutation/monomial representation of a group and to chop it into its irreducible constituents. For finite fields, the machinery for doing this has existed for some time—one can use Parker’s original Fortran-MeatAxe, M. Ringe’s translation of this into the C-MeatAxe or the versions of these on Magma. For infinite fields, one can use the characteristic 0 MeatAxe facilities on Magma or Parker’s integral MeatAxe [32]. There is also a standard method for splitting multiplicity-free permutation representations known as the ‘intersection matrix technique’. The degree of representation for which one can utilise the characteristic 0 MeatAxes is fairly limited for now.

One can always make new representations from old by tensoring two representations or taking symmetric/exterior powers of a given representation and then splitting the result.

6.2.3 Finding f_{H}(L) and f_{K}(L)

We can find generators for f_{H}(L) and f_{K}(L) using a ‘random’ search method. By changing our generators for f_{H}(L) and f_{K}(L), we must then ensure that f_{H}(L) = ⟨g_{1}, …, g_{m}⟩ and f_{K}(L) = ⟨h_{1}, … h_{n}⟩ where m = n and there is an isomorphism ϕ : f_{H}(L) → f_{K}(L) such that g_{i}ϕ = h_{i} for all i. (In the previous sentence, we shall usually have m = n = 2. Moreover, the generators for f_{H}(L) and f_{K}(L) will often be ‘standard’ generators.)

In our next step, we match up f_{H}(L) and f_{K}(L) by matching g_{i} and h_{i} for all i. But this matching determines the embeddings of L̄ = f_{H}(L) = f′_{K}(L) in f_{H}(H) and f′_{K}(K) and so we have to ensure that these embeddings are consistent with the amalgam we are trying to construct. If the amalgam H ⋆_{L} K is unique then we do not need to worry, otherwise we may have to apply an automorphism α ∈ Aut(f_{K}(L)) to f_{K}(L) before trying to match f_{H}(L) and f_{K}(L). Thus we must match g_{i} and h_{i}^{α} (for all i) for a suitable α ∈ Aut(f_{k}(L)).

6.2.4 The last two steps

To conjugate f_{K}(L) to f_{H}(L), we find a ‘canonical’ basis for the representation of L, which means that we can construct a ‘canonical’ (‘basis-independent’) representation of L. We then conjugate f_{K}(L) to f_{H}(L) via this ‘canonical’ representation. This process is described in some detail in Section 6.4 and is illustrated throughout this thesis. The ‘canonical’ basis is sometimes referred to as a ‘standard’ basis.

For the last step, we conjugate f′_{K}(K) to f′′_{K}(K) by an element of C_{GL(V)}(L̄). If we use standard basis techniques to put L̄ in block diagonal form, then it is easy to calculate C_{GL(V)}(L̄). In fact, the situation is not quite as bleak as this, we merely have to conjugate by (C_{GL(V)}(f′_{K}(K)), C_{GL(V)}(f_{H}(H)))-double coset representatives of C_{GL(V)}(L̄).

6.2.5 Checking the cases

If we are fortunate enough to have just one case to check then we are done. Otherwise, if F is finite then we have just a finite number of cases to check and for each case, we look at orders of ‘random’ elements of ⟨f_{H}(H), f′′_{K}(K)⟩, discarding those cases which contain element orders not found in G. This is usually enough to cut down the options to those cases where we have ⟨f_{H}(H), f′′_{K}(K)⟩ ≅ G.

For infinite fields, other methods must be found. We already know our representation up to conjugating f′_{K}(K) by elements of C_{GL(V)}(L̄). We can write a typical element of C_{GL(V)}(L̄) as a function of m̄ unknowns for some m̄, and we can write our typical (C_{GL(V)}(f′_{K}(K)), C_{GL(V)}(f_{H}(H)))-double coset representative as a function of m unknowns for some m ⩽ m̄. (It is easier to consider the whole centralising algebra of L̄ and worry about whether a particular algebra element has nonzero determinant later.) Our favourite method for determining these unknowns has been to use our knowledge of (expected) element traces (in ρ(G)) to introduce relations between them. If we are to engage in a construction by this technique, it helps to have another representation of G so that we can determine what class a particular element belongs to. A permutation representation or a representation of G over a finite field constructed by the technique outlined above will do for this purpose. In our case, a symmetric presentation of G will be available to us and this usually yields a manageable permutation representation of G.

6.2.6 How do we know we have the answer?

If we do know that G has a representation of the required type, then we can probably use a uniqueness argument to show that our matrices generate G. For example, this would happen if just one of the possible cases we were checking worked (i.e., was not immediately eliminated owing to some trivial consideration such as invalid element order/trace). There are variants on this case, such as two cases working and giving rise to dual representations of G.

If we do not know that G has a representation of the required type it can be quite tricky to show that the matrices we obtain do generate G. The method we adopt here is to find generators (for G) such that a presentation for G is known in terms of these generators. We then take words in these as generators for our representation of G. In our case, this means that we do not have to do anything since we shall be constructing our representations starting from symmetric presentations (see below, Section 6.3).

6.3 Our variant

It will probably not surprise the reader much that we intend to take G to be an image of T^{⋆n}:N and that we take H = N, K = T_{0}:N_{0} and L = H ∩ K = N_{0}. For simplicity of exposition, we shall assume that N = ⟨x, y⟩, N_{0} = ⟨x_{1}, y_{1}⟩, N^{0} = ⟨x_{2}, y_{2}⟩ and T_{0} = ⟨t⟩ ≅ C_{m} and the extra relations required to define G will be w_{i} = w_{i}(x, y, t) = 1 for i = 1, 2, etcetera. We shall let ρ, F and V be as in Section 6.2.

(i) Determine the restriction of ρ to H = N and construct this representation which we shall call ρ_{N}.

(ii) Use standard basis techniques (see Section 6.4) to find C_{GL(V)}(ρ_{N}(N^{0})). We have t ∈ C_{GL(V)}(ρ_{N}(N^{0})) and so we can regard t as a function in m unknowns where m is the dimension of the centralising algebra of ρ_{N}(N^{0}). In fact we need only consider those t which are not conjugate under conjugation by C_{GL(V)}(N).

(iii) Possibly reduce the number of unknowns by ensuring that N_{0} = ⟨x_{1}, y_{1}⟩ has

the correct normalising action on T_{0} and that t has the correct order.

(iv) Equate each of the w_{i} to 1. This gives relations between the unknowns.

(v) If ρ is (absolutely) irreducible and we took proper account of the conjugation action of C_{GL(V)}(N) on the t at Step (ii) we should be left with only a finite number of solutions for t, all of which should satisfy ⟨x, y, t⟩ ≅ G. If not, replace each ‘independent’ unknown that remains by 1 (or some other suitable value). We should now have ⟨x, y, t⟩ ≅ G. Check that x, y and t satisfy the presentation they were meant (and designed) to.

We do not need to elucidate the above points further since we have already covered them in Section 6.2. However, we point out that it is useful to consider element traces to produce (linear) dependencies between the unknowns before we embark in any serious manner on Steps (iii)–(v) above. (Steps (iii) and (iv) need not be done in the order listed. Also, we have often neglected to consider the equivalence of candidate t under the conjugation action of N until quite late in the construction.) For the purposes of identifying the conjugacy class in which a particular element lies, we shall usually utilise a permutation representation of G that is obtained by coset enumeration (of G over a subgroup) using a presentation of G, which in turn is obtained from the symmetric presentation of G we are considering.

When we compare our method to the general amalgam method some disadvantages are apparent, for the amalgams (and possibly the representations also) which can be dealt with by our method are a subset of those which can be dealt with by the general method. Also, we must find a suitable symmetric presentation of G. However, there are some advantages, for example, on our way to constructing G we only construct a representation of H = N and never (explicitly) construct a representation of K = T_{0}:N_{0}. More importantly, we do not need to worry that our embeddings of L in H and K are consistent with the amalgam we are trying to construct—the presentation takes care of this for us. Moreover, after making our representation, it is easy to verify that we have a representation of G—we just verify that the matrices of x, y and t satisfy the relations of the presentation of G.

6.3.1 An example—a 111-dimensional representation of Ly

The example given by Parker and Wilson [33] is suitable for our purposes. This example is a 111-dimensional (orthogonal) representation of the Lyons group Ly over \overline{𝔽}_{5}, where we consider Ly as a homomorphic image of the monomial progenitor 3^{⋆2520}:_{m} 2·A_{11}. (In fact, the representation turns out to be writable—and constructed—over 𝔽_{5}.) We do not yet possess a symmetric presentation of Ly as a homomorphic image of 3^{⋆2520}:_{m} 2·A_{11} in a form that is useful for constructing Ly and verifying that the matrices we eventually obtain generate Ly. However, in the case of the construction, there turns out to be just one case to consider anyway. We shall ignore the problem of verification.

It is clear from the ‘modular Atlas’ [27] and the partial putative Brauer character given in [33] that the only possible restriction of our (putative) representation to the control subgroup N ≅ 2·A_{11} is as a direct sum 55a ⊕ 56a/b where the 55a is a representation of A_{11} and the 56a/b is faithful. We must now construct these representations, both of which were already available on the World Wide Web in Wilson’s library [40]. Constructing the representations of N is meant to be easy—but that is not so here. We have deferred the construction of the representations of N till the end of this (sub-)section.

The restrictions of the representations 55a, 56a and 56b of N ≅ 2·A_{11} to N^{0} ≅ M_{11} are as the direct sums 10a ⊕ 45a, 11a ⊕ 45a and 1a ⊕ 55a respectively—in each case, the two summands are in different blocks. Thus the centraliser of N^{0} is easy to write down. If the restriction to N is as 55a ⊕ 56b then any element t that commutes with N^{0} must preserve the block structure of N and thus the restriction to ⟨N, t⟩ = G cannot be irreducible. In any case, there can be no elements t of order 3 that commute with N^{0} and are inverted by the central involution of N.

Thus the restrictions to N ≅ 2·A_{11} and N^{0} ≅ M_{11} respectively are as 55a ⊕ 56a and (10a⊕45a)⊕(11a⊕45a). If we have the ‘wrong’ faithful 56-dimensional representation of N then we apply an outer automorphism of N to obtain the ‘correct’ one.^{1} We shall let z be the central involution of N and let t be any element of C_{GL(V)}(N^{0}), where V = \overline{𝔽}_{5}^{111}, such that t has order 3 and is inverted by z. Note that N_{0} = N^{0} × ⟨z⟩ ≅ M_{11} × 2. Our elements/representations take the following forms with respect to some standardised basis:

N ∼ [ \begin{matrix}55a \\ & 56a \end{matrix} ] , N^{0} ∼ [ \begin{matrix}10a \\ & 45a \\ & & 11a \\ & & & 45a \end{matrix} ] , t ∼ [ \begin{matrix}ϵI_{10} \\ & αI_{45} & & βI_{45} \\ & & ζI_{11} \\ & γI_{45} & & δI_{45} \end{matrix} ] ,

where α, β, γ, δ, ϵ and ζ are elements of \overline{𝔽}_{5}, I_{n} denotes the n × n identity matrix and the blank areas consist solely of zeros (and the representations of N and N^{0} are written over 𝔽_{5}). The conditions that t^{3} = 1 ≠ t and t^{z} = t^{−1} force ϵ = ζ = 1, α = δ = −1/2 and βγ = −3/4. (Some of these calculations can be simplified by observing that, since the representation is irreducible, β and γ are nonzero.) The remaining possibilities are conjugate under C_{GL(V)}(N) and must give rise to isomorphic groups. Working modulo 5, we choose ϵ = ζ = 1, α = δ = 2, β = 1 and γ = 3. This must be a representation of the Lyons group if it has one of this degree, which we know it does.

Constructing the representations of N ≅ 2·A_{11}

We may construct the 55-dimensional 𝔽_{5}-representation of A_{11} fairly easily. Take the symmetric square of the 11-dimensional permutation representation and chop out the 43-dimensional constituent, then take the symmetric square of this and chop out the (required) 55-dimensional constituent.

Constructing the 56-dimensional representation of 2·A_{11} is harder. We may take:

⟨x, y, z | x^{3} = y^{9} = (xy)^{11} = z^{2} = [y, z] = 1, (xx^{y})^{2} = (xx^{y^{2}})^{2} = (xx^{y^{3}})^{2} = (xx^{y^{4}})^{2} = z⟩,

as a presentation for N ≅ 2·A_{11}. (The relation [x, z] = 1 also holds.) We then construct a faithful degree 5040 permutation representation of N over ⟨xy, zyxy^{4}x^{−1}y^{2}xy^{−3}⟩ ≅ M_{11} using coset enumeration. The required 56-dimensional representation is a constituent of this permutation representation reduced modulo 5. However, chopping this representation took some 3\frac{1}{4} hours CPU time when executed in Magma on fourier.

It is better to heed the suggestion of Parker and Wilson [33]. A minor variant is to reduce the 16-dimensional representation of 2·A_{11} over ℤ[b11] modulo 5, tensor with the natural permutation representation of A_{11} and chop out the required 56-dimensional representation.^{2} Since we did not have a 16-dimensional ℤ[b11]-representation of 2·A_{11} to hand we decided to make one. This turned out not to be too difficult, taking as our starting point an 8-dimensional representation of 2·A_{9} which we can obtain by restricting an 8-dimensional representation of 2·O_{8}^{+}(2):2, the Weyl group of E_{8}.

1Parker and Wilson [33] fixed the 56-dimensional representation of 2·A_{11} and changed the M_{11} if necessary whereas we fix the M_{11} and change the 56-dimensional representation if necessary.

2Parker and Wilson started from the 32-dimensional representation of 2·A_{11} over ℤ.

6.4 Finding a canonical basis

6.4.1 Create seed—CSeed

Given a representation, we must be able to specify up to scalar multiplication a nonzero vector in our space V = F^{n}. The MeatAxe accomplishes this by seeking an element of the group algebra F G with nullity 1 and taking a nonzero element in the null-space of this element. Perhaps we could use that method here. Seed vectors for representations should always be understood to be nonzero even if we do not explicitly say so.

We have opted to seed representations by fixed vectors of subgroups. As a variant, we have also seeded representations by vectors that are in certain non-trivial 1-dimensional submodules of subgroups. As an example, we take the representations of N ≅ 2·A_{11} and N_{0} ≅ M_{11} above, where we must also split the restrictions of the representations of N to N_{0} into irreducible components. We take as our seed vector of representation 10a of M_{11} any fixed vector of (a particular) M_{10}. We seed 11a (of M_{11}) with any fixed vector of L_{2}(11). Alternatively, we could seed 11a with any vector fixed by A_{6} and negated by outer elements of M_{10}. Our proposed seed for 45a is any vector fixed by 3^{2}:Q_{8} and negated by outer elements of 3^{2}:(8:_{3}2). However, there are such vectors in 10a. Therefore we take any vector that is fixed by 2·A_{4} and negated by outer elements of 2·S_{4}^{+} and ‘spin up’ to the 45-space. Once we have found the 45-space, we seed it as we intended.

It is particularly difficult to pick seed vectors for representations with Schur index greater than 1. (A representation has Schur index 1 just when it can be written over the field generated by its character values.) In characteristic zero, all representations with indicator −1, including many faithful representations of double covers, have even Schur index—in particular, a Schur index > 1.

6.4.2 Standard basis—SBasis

We wish to produce a standard basis {v_{1}, v_{2}, …, v_{n}} for our representation, which we will assume to be irreducible. We take v(≠ 0) to be a seed vector produced using a CSeed procedure. Let G = ⟨x, y⟩. Then we consider the following list of vectors:

v, vx, vy, vxx, vxy, vyx, vyy, vxxx, vxxy, vxyx, vxyy, … etc.,

and define our standard basis {v_{1}, v_{2}, …, v_{n}} to be those vectors in the list which are linearly independent of all their predecessors. This performs the same function as the MeatAxe command zsb. We shall sometimes refer to this process as spinning up the seed vector v = v_{1} under the action of G.

Apart from its use for standardising a basis, SBasis has other uses. One of these— finding invariant sesquilinear forms—is described below. The other use is to write a representation over a minimal field, but this requires a suitable choice of seed vector— particularly difficult for representations with Schur index greater than 1. (Notice the use of i, where i^{2} = 1, in defining the seed vector for the 4-dimensional representation of 2·A_{5} in Section 6.4.4.)

6.4.3 Finding forms

For various reasons it is useful to know if a given representation leaves a sesquilinear form invariant. If it does, then we should quite like to find it. We describe how to find an invariant sesquilinear form below. We have made use of these techniques in Section 7.7 where we determine all images of a given progenitor that are L_{2}(q) or PGL_{2}(q). We need these techniques because we construct a 3-dimensional representation of the progenitor we are investigating and make use of the isomorphism PGL_{2}(q) ≅ SO_{3}(q). This means that we have to find the symmetric bilinear (that is, orthogonal) form preserved by the representation.

Throughout, we shall assume that we are dealing with a representation of ρ : G = ⟨x, y⟩ → GL(V) where F is a field and V = F^{n}. Moreover, in the Hermitian case, we shall assume that σ is an automorphism of F of order 2. For convenience, we shall assume that our representation is absolutely irreducible. We have divided our analysis into two parts according as the representation is self-dual or not.

Symplectic and orthogonal forms—the self-dual case

Let ρ^{∗} be the dual of ρ—so ρ and ρ^{∗} are equivalent. Then for all x ∈ G we have ρ^{∗}(x) = ρ(x)^{−T}, the inverse transpose of ρ(x). Use the techniques described in Sections 6.4.2 and 6.4.1 to find a standard basis {v_{1}, v_{2}, …, v_{n}} for ρ(G). Use the same procedure to obtain a standard basis {w_{1}, w_{2}, …, w_{n}} for ρ^{∗}(G). Define L_{1} and L_{2} to be the matrices whose i^{th} rows are v_{i} and w_{i} respectively. Thus we have

L_{1} = [ v_{1} ; v_{2} ; ⋮ ; v_{n} ] and L_{2} = [ w_{1} ; w_{2} ; ⋮ ; w_{n} ] .

Then L_{1}^{−1} conjugates ρ(G) to a standard form and L_{2}^{−1} conjugates ρ^{∗}(G) to the same standard form. Let B = L_{1}^{−1}L_{2}. Then B conjugates ρ(G) to ρ^{∗}(G). Writing x for ρ(x), we find that for all x ∈ G we have:

x^{B} = x^{−T} ⟺ B^{−1}xBx^{T} = 1
⟺ xBx^{T} = B
⟺ (xBx^{T})^{T} = B^{T}
⟺ xB^{T}x^{T} = B^{T}.

But the matrix B that conjugates ρ(G) to ρ^{∗}(G) is unique up to scalar multiplication since we assumed that ρ was absolutely irreducible. Therefore B^{T} = λB for some λ ∈ F and so B = (B^{T})^{T} = (λB)^{T} = λB^{T} = λ^{2}B. Thus λ^{2} = 1 and so B^{T} = B or −B and hence B is the required orthogonal/symplectic form. Note that for orthogonal groups in characteristic 2, this technique finds the symplectic form preserved by ρ(G). Further work is then required to find the quadratic form that ρ(G) leaves invariant.

Hermitian (or unitary) forms—the not necessarily self-dual case

Here we assume that ρ^{σ}, defined by ρ^{σ}(x) = ρ(x)^{σ} for all x ∈ G, is equivalent to the dual of ρ. Equivalently, we assume that ρ̂, defined by ρ̂(x) = ρ(x)^{−Tσ} for all x ∈ G, is equivalent to ρ.

Proceed as in the self-dual case to create matrices L_{1} and L_{2} whose inverses conjugate ρ(G) and ρ̂(G) respectively into standard form. Let B̃ = L_{1}^{−1}L_{2}. Then B̃ conjugates ρ to ρ̂. Writing x for ρ(x) we find that for all x ∈ G we have:

xB̃ = x^{−Tσ} ⟺ B̃^{−1}xB̃x^{Tσ} = 1
⟺ xB̃x^{Tσ} = B̃
⟺ (xB̃x^{Tσ})^{Tσ} = B̃^{Tσ}
⟺ xB̃^{Tσ}x^{Tσ} = B̃^{Tσ}.

By uniqueness we get B̃^{Tσ} = λB̃ for some λ ∈ F and hence B̃ = (B̃^{Tσ})^{Tσ} = λλ^{σ}B which gives λλ^{σ} = 1. Now let B = κB̃. Then B is the matrix of a (non-singular) σ-Hermitian form just when κ = κ^{σ}λ, that is, B^{Tσ} = B just when κ/κ^{σ} = λ. If σ is complex conjugation or if σ is an automorphism of order 2 of a finite field then we can always solve κ/κ^{σ} = λ for κ ∈ F. Indeed, in such cases, it can usually be done by inspection of the diagonal entries of B.

6.4.4 An example—a 4-dimensional representation of 2·A_{5}

We illustrate the above methods with the 4-dimensional, faithful, irreducible representation of 2·A_{5} over ℂ (or rather ℤ[i] where i^{2} = −1). This representation is equivalent to its dual. It is generated by:

x_{1} ∼ [ \begin{matrix}−i & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 2 − i & 1 + i & i & −1 + i \\ 0 & −1 & 0 & 0 \end{matrix} ] and y_{1} ∼ [ \begin{matrix}0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 1 & 0 & 0 & 0 \\ −2 + i & −2 − i & −2i & 1 \end{matrix} ] ,

where x_{1}^{4} = y_{1}^{3} = (x_{1}y_{1})^{5} = [x_{1}^{2}, y_{1}] = 1 and x_{1}^{2} = −1 (or −I_{4}). We shall let x_{2} = x_{1}^{−T}, y_{2} = y_{1}^{−T}, x_{3} = x_{1}^{−Tσ} and y_{3} = y_{1}^{−Tσ} where x^{−T} is the inverse transpose of x and σ is complex conjugation. The representations of 2·A_{5} generated by G_{i} = ⟨x_{i}, y_{i}⟩ for each i are all equivalent. For i ∈ {1, 2, 3}, we take our seed vector v_{i} to be any nonzero vector such that v_{i}x_{i} = iv_{i} and v_{i}x_{i}^{y_{i}x_{i}y_{i}} = −iv_{i}. This defines v_{i} up to scalar multiplication. We then take {v_{i}, v_{i}y_{i}, v_{i}y_{i}x_{i}, v_{i}y_{i}^{2}} as our standard basis and define L_{i} to be the matrix whose rows are vectors of the standard basis. Thus we obtain:

L_{i} = [ \begin{matrix}v_{i} \\ v_{i}y_{i} \\ v_{i}y_{i}x_{i} \\ v_{i}y_{i}^{2} \end{matrix} ] , giving L_{1} = \frac{1}{5} [ \begin{matrix}5 & 3 + 4i & −2 + 4i & −2 − i \\ 3 + 4i & −2 + 4i & −5 & −2 − i \\ −6 + 2i & −3 − 4i & −5i & 3 − i \\ 0 & 0 & 0 & −2 − i \end{matrix} ] ,

along with

L_{2} = \frac{1}{4} [ \begin{matrix}4 & 1 + i & 4i & 1 − i \\ 4i & −4 & −1 − i & −1 + i \\ −4 & 1 − i & −1 − i & −4 \\ −1 − i & −4i & 4 & 0 \end{matrix} ] and L_{3} = [ \begin{matrix}0 & 1 & 0 & −i \\ 0 & 0 & −1 & i \\ 0 & −i & −1 & 0 \\ −1 & 0 & 0 & −2 \end{matrix} ] .

Conjugating G_{i} by L_{i}^{−1} puts the representation into standard form. Thus

B_{S} = \frac{1}{5}(−2 − 6i)L_{1}^{−1}L_{2} = [ \begin{matrix}0 & 1 & 1 & i \\ −1 & 0 & 1 & −2 + 2i \\ −1 & −1 & 0 & −2 − 2i \\ −i & 2 − 2i & 2 + 2i & 0 \end{matrix} ] ,

conjugates G_{1} to G_{2} and is a non-singular symplectic/orthogonal (in this case symplectic) form preserved by G_{1}. The determinant of B_{S} is −9. (In fact, L_{1}^{−1}L_{2} also conjugates G_{1} to G_{2} and is a non-singular symplectic form preserved by G_{1}, but the form given above is neater.) Similarly, we find that

B_{I} = L_{1}^{−1}L_{3} = [ \begin{matrix}4 − 2i & −1 − 2i & −1 − 2i & 2 − i \\ 1 + 2i & 4 − 2i & −1 − 2i & 0 \\ 1 + 2i & 1 + 2i & 4 − 2i & 0 \\ 2 − i & 0 & 0 & 4 − 2i \end{matrix} ] ,

conjugates G_{1} to G_{3} but it is not a non-singular Hermitian form—just a scalar multiple of one. We find that

B_{U} = \frac{1}{2 − i} B_{I} = [ \begin{matrix}2 & −i & −i & 1 \\ i & 2 & −i & 0 \\ i & i & 2 & 0 \\ 1 & 0 & 0 & 2 \end{matrix} ] ,

is (the matrix of) a non-singular Hermitian form preserved by G_{1}. The determinant of B_{U} is 1. We conclude this section by giving generators for the representation with respect to the standardised basis. We show these below:

x_{s} = [ \begin{matrix}i & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \\ \frac{1}{2}(3 + i) & \frac{1}{2}(−1 + i) & \frac{1}{2}(1 + i) & −i \end{matrix} ] and y_{s} = [ \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ i & −1 − i & 1 & 1 \\ 1 & 0 & 0 & 0 \end{matrix} ] ,

where x_{s} = x_{i}^{L_{i}^{−1}} and y_{s} = y_{i}^{L_{i}^{−1}} for all i.

The systematic approach

7.1 Introduction

In my M. Phil. [3], a systematic search for (small) groups generated by small, but highly symmetric, sets of involutions or elements of order 3 was carried out. Since we do not expect that document to get published (except those parts of it that appear in the papers [19, 4]), we have decided to give a brief summary of the results here. In places, we improve on the results given in Bray [3].

For symmetric generators of order 2, we studied the progenitors 2^{⋆3}:S_{3}, 2^{⋆4}:A_{4}, 2^{⋆4}:S_{4}, 2^{⋆5}:A_{5}, 2^{⋆6}:L_{2}(5), 2^{⋆5}:S_{5}, 2^{⋆7}:L_{3}(2), 2^{⋆8}:L_{2}(7) and 2^{⋆8}:PGL_{2}(7). For symmetric generators of order 3, we studied the permutation progenitors 3^{⋆3}:S_{3}, 3^{⋆4}:A_{4}, 3^{⋆4}:S_{4}, 3^{⋆5}:A_{5}, 3^{⋆6}:L_{2}(5), 3^{⋆5}:S_{5}, 3^{⋆7}:L_{3}(2), 3^{⋆8}:L_{2}(7) and 3^{⋆8}:PGL_{2}(7) and the monomial progenitors 3^{⋆3}:_{m} S_{3}, 3^{⋆4}:_{m} S_{4}, one case of 3^{⋆3}:_{m} S_{4}, 3^{⋆4}:_{m} 2·S_{4}^{+}, 3^{⋆6}:_{m} L_{2}(5), 3^{⋆5}:_{m} S_{5} and 3^{⋆7}:_{m} L_{3}(2). We also briefly considered the modular group 2^{⋆3}:A_{3} ≅ 3^{⋆2}:S_{2} ≅ 3^{⋆2}:_{m} S_{2} ≅ 2 ⋆ 3 ≅ L_{2}(ℤ). In addition, we shall study the progenitor 3^{⋆2}:_{m} D_{8}. This progenitor is isomorphic to 2^{⋆3}:S_{3} extended by its outer automorphism, indeed, we have the isomorphism 3^{⋆2}:_{m} 2^{2} ≅ 2^{⋆3}:S_{3}.

7.2 Presentations of our progenitors

These are easy to obtain from the formula:

m^{⋆n}:N = ⟨x, y, t | ⟨x, y⟩ ≅ N, t^{m} = 1 = [t, N^{0}]⟩,

for a permutation progenitor, where [t, N^{0}] = 1 means that we adjoin sufficient relations to ensure that N^{0} and t = t_{0} commute. In the monomial case, the corresponding presentation is:

m^{⋆n}:_{m} N = ⟨x, y, t | ⟨x, y⟩ ≅ N, t^{m} = 1, ⟨t⟩^{N_{0}} = ⟨t⟩⟩,

where ⟨t⟩^{N_{0}} = ⟨t⟩ means that we adjoin enough relations to specify the normalising action of N_{0} on T_{0} = ⟨t⟩.

We reproduce the presentations of all our progenitors below (in Tables 7.1, 7.2 and 7.3). We also give the actions of x and y on the symmetric generators, as well as which symmetric generator t is. For the monomial progenitors with symmetric generators of order 3, ī denotes t_{i}^{−1}.

7.3 Relations we append to our progenitors

7.3.1 n^{th} order parameters

In our systematic search, the relations that we append to our progenitors take one of two forms. The first form is specifying the orders of (some of) the o(πt_{i}^{ε}) for π ∈ N and i = 0, …, n−1. These are the first order relations. As an extension of this, for the ‘smaller’ progenitors, we shall occasionally use second order and even third order relations, i.e., relations of the types o(πt_{i}^{ε_{1}}t_{j}^{ε_{2}}) = n and o(πt_{i}^{ε_{1}}t_{j}^{ε_{2}}t_{k}^{ε_{3}}) = m. All the progenitors we consider are sufficiently ‘nice’ so as to produce just a few first order relations (which are easily calculated) and so we may parametrise our progenitors in terms of all the first order relations (and perhaps some other well-chosen relations).

7.3.2 s-type relations

The second form of relation we consider derives from Curtis’s Lemma that

⟨t_{i}, t_{j}⟩ ∩ N ⩽ C_{N}(N^{ij}).

In each of the monomial progenitors except 3^{⋆5}:_{m} S_{5}, this produces no restriction—indeed, one can have G = ⟨t_{0}, t_{1}⟩ in those cases. In 3^{⋆5}:_{m} S_{5}, we have ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(2, 3, 4)⟩.

For the permutation progenitors m^{⋆n}:N, Curtis’s Lemma (Lemma 3.6.2) gives the following restrictions:

n  N  Restriction on ⟨t_{i}, t_{j}⟩ ∩ N
3  S_{3}  None
4  A_{4}  None
4  S_{4}  ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(0, 1), (2, 3)⟩
5  A_{5}  ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(2, 3, 4)⟩
6 L_{2}(5) ⟨t_{∞}, t_{0}⟩ ∩ N ⩽ ⟨(∞, 0)(1, 4), (1, 4)(2, 3)⟩
5 S_{5} ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(0, 1)⟩
7 L_{3}(2) ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(2, 5)(4, 6), (2, 6)(4, 5)⟩
8 L_{2}(7) ⟨t_{∞}, t_{0}⟩ ∩ N ⩽ ⟨(1, 2, 4)(3, 6, 5)⟩
8 PGL_{2}(7) ⟨t_{∞}, t_{0}⟩ ∩ N ⩽ ⟨(1, 3, 2, 6, 4, 5)⟩

For the case m = 3 we can say no more than this, but in the case m = 2, we make use of the fact that ⟨t_{i}, t_{j}⟩ is dihedral to obtain further restrictions.

We can use the rather easy-to-prove fact that the centre of a dihedral group (other than D_{4} ≅ 2^{2}) has order at most 2 to show that any element of ⟨t_{i}, t_{i}⟩ ∩ N that fixes i and j must be the identity or have order 2. We may use some general, and less trivial, results from Section 3.6.2 to obtain further restrictions. For example, if |G : N| > 2, then we do

Table 7.1: Presentations of the progenitors 2^{⋆n}:N that we are considering

(i)  ⟨x, t | x^{3} = t^{2} = 1⟩ ≅ 2^{⋆3}:A_{3} ≅ 2 ⋆ 3
x ∼ (0, 1, 2), t ∼ t_{0}.
(ii)  ⟨x, y, t | x^{3} = y^{2} = (yx)^{2} = 1 = t^{2} = [t, y]⟩ ≅ 2^{⋆3}:S_{3}
x ∼ (0, 1, 2), y ∼ (1, 2), t ∼ t_{0}.
(iii)  ⟨x, y, t | x^{3} = y^{3} = (xy)^{2} = 1 = t^{2} = [x, t]⟩ ≅ 2^{⋆4}:A_{4}
x ∼ (1, 2, 3), y ∼ (0, 1, 2), t ∼ t_{0}.
(iv)  ⟨x, y, t | x^{4} = y^{2} = (yx)^{3} = 1 = t^{2} = [y, t] = [y, t^{x}]⟩ ≅ 2^{⋆4}:S_{4}
x ∼ (0, 1, 2, 3), y ∼ (2, 3), t ∼ t_{0}.
(v)  ⟨x, y, t | x^{5} = y^{3} = (xy)^{2} = 1 = t^{2} = [t, y] = [t, y^{x^{2}}]⟩ ≅ 2^{⋆5}:A_{5}
x ∼ (0, 1, 2, 3, 4), y ∼ (4, 2, 1), t ∼ t_{0}.
(vi)  ⟨x, y, t | x^{5} = y^{3} = (xy)^{2} = 1 = t^{2} = [t, x] = [t^{yx^{2}}, xy]⟩ ≅ 2^{⋆6}:L_{2}(5)
x ∼ (0, 1, 2, 3, 4), y ∼ (∞, 0, 1)(2, 4, 3), t ∼ t_{∞}.
(vii) ⟨x, y, t | x^{5} = y^{2} = (xy)^{4} = [x, y]^{3} = 1 = t^{2} = [t, y] = [t^{x}, y] = [t^{x^{2}}, y]⟩ ≅ 2^{⋆5}:S_{5}
x ∼ (0, 1, 2, 3, 4), y ∼ (3, 4), t ∼ t_{0}.
(viii)  ⟨x, y, t | x^{7} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{2} = [t^{x^{4}}, xy] = [t, y^{x^{4}}](= [t, y])⟩

≅ 2^{⋆7}:L_{3}(2)

x ∼ (0, 1, 2, 3, 4, 5, 6), y ∼ (2, 6)(4, 5), t ∼ t_{0}.
(ix) ⟨x, y, t | x^{7} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{2} = [x, t] = [t^{x^{2}}, xy]⟩ ≅ 2^{⋆8}:L_{2}(7)
x ∼ (0, 1, 2, 3, 4, 5, 6), y ∼ (∞, 0)(1, 6)(2, 3)(4, 5), t ∼ t_{∞}.
(x) ⟨x, y, t | x^{8} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{2} = [t, y](= [t^{x^{−2}}, y]) = [t^{x^{−1}}, x^{4}y]⟩

≅ 2^{⋆8}:PGL_{2}(7)

x ∼ (∞, 2, 5, 4, 6, 1, 0, 3), y ∼ (1, 6)(2, 5)(3, 4), t ∼ t_{∞}.

(xi)  ⟨x, y, t | x^{3} = y^{2} = (yx)^{2} = 1 = t^{3} = [t, y]⟩ ≅ 3^{⋆3}:S_{3}
x ∼ (0, 1, 2), y ∼ (1, 2), t ∼ t_{0}.
(xii)  ⟨x, y, t | x^{3} = y^{3} = (xy)^{2} = 1 = t^{3} = [x, t]⟩ ≅ 3^{⋆4}:A_{4}
x ∼ (1, 2, 3), y ∼ (0, 1, 2), t ∼ t_{0}.
(xiii)  ⟨x, y, t | x^{4} = y^{2} = (yx)^{3} = 1 = t^{3} = [t, y] = [t^{x}, y]⟩ ≅ 3^{⋆4}:S_{4}
x ∼ (0, 1, 2, 3), y ∼ (2, 3), t ∼ t_{0}.
(xiv)  ⟨x, y, t | x^{5} = y^{3} = (xy)^{2} = 1 = t^{3} = [t, y] = [t, y^{x^{2}}]⟩ ≅ 3^{⋆5}:A_{5}
x ∼ (0, 1, 2, 3, 4), y ∼ (4, 2, 1), t ∼ t_{0}.
(xv) ⟨x, y, t | x^{5} = y^{2} = (yx)^{4} = [x, y]^{3} = 1 = t^{3} = [t, y] = [t^{x}, y] = [t^{x^{2}}, y]⟩ ≅ 3^{⋆5}:S_{5}
x ∼ (0, 1, 2, 3, 4), y ∼ (3, 4), t ∼ t_{0}.
(xvi)  ⟨x, y, t | x^{5} = y^{3} = (xy)^{2} = 1 = t^{3} = [t, x] = [t^{yx^{2}}, xy]⟩ ≅ 3^{⋆6}:L_{2}(5)
x ∼ (0, 1, 2, 3, 4), y ∼ (∞, 0, 1)(2, 4, 3), t ∼ t_{∞}.
(xvii) ⟨x, y, t | x^{7} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{3} = [t^{x^{4}}, xy] = [t, y^{x^{4}}]⟩ ≅ 3^{⋆7}:L_{3}(2)
x ∼ (0, 1, 2, 3, 4, 5, 6), y ∼ (2, 6)(4, 5), t ∼ t_{0}.
(xviii) ⟨x, y, t | x^{7} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{2} = [x, t] = [t^{x^{2}}, xy]⟩ ≅ 3^{⋆8}:L_{2}(7)
x ∼ (0, 1, 2, 3, 4, 5, 6), y ∼ (∞, 0)(1, 6)(2, 3)(4, 5), t ∼ t_{∞}.
(xix) ⟨x, y, t | x^{8} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{2} = [t, y] = [t, x^{3}yx]⟩ ≅ 3^{⋆8}:PGL_{2}(7)
x ∼ (∞, 2, 5, 4, 6, 1, 0, 3), y ∼ (1, 6)(2, 5)(3, 4), t ∼ t_{∞}.

Table 7.3: Presentations of the progenitors 3^{⋆n}:_{m} N that we are considering

(xx)  ⟨x, y, t | x^{4} = y^{2} = (yx)^{2} = 1 = t^{3} = [t, y] = (x^{2}t)^{2}⟩ ≅ 3^{⋆2}:_{m} D_{8}
x ∼ (0, 1, ¯0, ¯1), y ∼ (1, ¯1), t ∼ t_{0}.
(xxi)  ⟨x, y, t | x^{3} = y^{2} = (yx)^{2} = 1 = t^{3} = (yt)^{2}⟩ ≅ 3^{⋆3}:_{m} S_{3}
x ∼ (0, 1, 2), y ∼ (0, ¯0)(1, ¯2), t ∼ t_{0}.
(xxii)  ⟨x, y, t | x^{4} = y^{2} = (yx)^{3} = 1 = t^{3} = (yt)^{2} = (yt^{x})^{2}⟩ ≅ 3^{⋆4}:_{m} S_{4}
x ∼ (0, ¯1, 2, ¯3), y ∼ (0, ¯0)(1, ¯1)(2, ¯3), t ∼ t_{0}.
(xxiii)  ⟨x, y, t | x^{4} = y^{2} = (yx)^{3} = 1 = t^{3} = [t, y] = (x^{2}t)^{2}⟩ ≅ 3^{⋆3}:_{m} S_{4}
x ∼ (0, 1, ¯0, ¯1)(2, ¯2), y ∼ (1, ¯2), t ∼ t_{0}.
(xxiv)  ⟨x, y, t | x^{4} = (yx)^{3}, y^{2} = 1 = t^{3} = [t, y] = (yt^{x})^{2}⟩ ≅ 3^{⋆4}:_{m} 2·S_{4}^{+}
x ∼ (0, ¯1, ¯2, 3, ¯0, 1, 2, ¯3), y ∼ (1, ¯1)(2, 3), t ∼ t_{0}.
(xxv) ⟨x, y, t | x^{5} = y^{3} = (xy)^{2} = 1 = t^{3} = [t, x] = (t^{yx^{2}}xy)^{2}⟩ ≅ 3^{⋆6}:_{m} L_{2}(5)
x ∼ (0, 1, 2, 3, 4), y ∼ (∞, 0, 1)(2, 4, ¯3), t ∼ t_{∞}.
(xxvi) ⟨x, y, t | x^{5} = y^{2} = (yx)^{4} = [x, y]^{3} = 1 = t^{3} = (yt)^{2} = (yt^{x})^{2} = (yt^{x^{2}})^{2}⟩

≅ 3^{⋆5}:_{m} S_{5}

x ∼ (0, 1, 2, 3, 4), y ∼ (0, ¯0)(1, ¯1)(2, ¯2)(3, 4), t ∼ t_{0}.
(xxvii) ⟨x, y, t | x^{7} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{3} = [t^{x^{4}}, xy] = (ty^{x^{4}})^{2}(= [t, y])⟩

≅ 3^{⋆7}:_{m} L_{3}(2)

x ∼ (0, 1, 2, 3, 4, 5, 6), y ∼ (1, ¯1)(3, ¯3)(2, ¯6)(4, 5), t ∼ t_{0}.

obtain some (proper) restriction on ⟨t_{i}, t_{i}⟩ ∩ N. We summarise the revised results in the table below.

Progenitor Restriction on ⟨t_{i}, t_{j}⟩ ∩ N
2^{⋆3}:A_{3}  ⟨t_{0}, t_{1}⟩ ∩ N = 1 if |G : N| > 2.
2^{⋆3}:S_{3}  ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(0, 1)⟩ if |G : N| > 2.
2^{⋆4}:A_{4}  ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(0, 1)(2, 3)⟩ if |G : N| > 2.
2^{⋆4}:S_{4}  ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(0, 1), (2, 3)⟩
2^{⋆5}:A_{5}  ⟨t_{0}, t_{1}⟩ ∩ N = 1
2^{⋆6}:L_{2}(5) ⟨t_{∞}, t_{0}⟩ ∩ N ⩽ ⟨(∞, 0)(1, 4), (1, 4)(2, 3)⟩
2^{⋆5}:S_{5}  ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(0, 1)⟩
2^{⋆7}:L_{3}(2) ⟨t_{0}, t_{1}⟩ ∩ N ⩽ ⟨(2, 5)(4, 6)⟩
2^{⋆8}:L_{2}(7) ⟨t_{∞}, t_{0}⟩ ∩ N = 1
2^{⋆8}:PGL_{2}(7) ⟨t_{∞}, t_{0}⟩ ∩ N ⩽ ⟨(1, 6)(2, 5)(3, 4)⟩

Thus, for the progenitors 2^{⋆n}:N that we are considering (and many more such progenitors), we have that ⟨t_{i}, t_{j}⟩∩N ⩽ M, where each element of M either fixes both i and j or swaps them—at least when |G : N| > 2. If [t_{i}t_{j}]^{k}t_{i} = π ∈ M, then if π fixes i and j we get collapse (to π = t_{i} or t_{j}) and so π, of order 2, must swap i and j and so this relation becomes [πt_{i}]^{2k+1} = 1, a first order relation. If [t_{i}t_{j}]^{k} = π ∈ M then if π exchanges i and j then this relation collapses (to π = 1) and so π must fix both i and j. If π = 1 then this is the second order relation [t_{i}t_{j}]^{k} = 1 which, if there is an involution π_{ij} that swaps i and j, is equivalent to the first order relation [t_{i}π_{ij}]^{2k} = 1. But if π ≠ 1 then this relation (that is, [t_{i}t_{j}]^{k} = π) is a proper s-type relation. If k is odd then this may be expressed as the second order relation [πt_{i}t_{j}]^{k} = 1.

For permutation progenitors with generators of order 3, we have proceeded by analogy with the involutory generators case and included s-type parameters (for the progenitor with generators of order 3) just when the ‘corresponding’ progenitor with involutory generators had them. Even then we have only included the ‘nicer’ looking relations. For example, in 2^{⋆4}:S_{4} we have the (possible) relation (2, 3) = [t_{0}t_{1}]^{s}, so in 3^{⋆4}:S_{4} we decided to include the possible relations (2, 3) = [t_{0}t_{1}]^{s_{1}} and (2, 3) = [t_{0}t_{1}^{−1}]^{s_{2}}. Of course, in the latter case, there are many more relations than this.

7.3.3 Historical note

Our first computer implementation of procedures for finding presentations of finite homomorphic images of the progenitors given here was on Cayley which is not case-sensitive. This meant that we used g, n, h, k and l for groups, x, y and t for group elements and a, b, c, d, e, f, p, q, r etc., for the n^{th} order parameters and s, s_{1} and s_{2} for the s-type parameters.

7.3.4 The actual cases—involutory generators

Table 7.4 gives all the relations used in the progenitors 2^{⋆n}:N we are considering except 2^{⋆3}:A_{3} which we have considered below. These include all first order relations as well as the relations of s-type when applicable. Only 2^{⋆4}:S_{4}, 2^{⋆6}:L_{2}(5) and 2^{⋆8}:PGL_{2}(7) have s-type relations. We have included all second order relations in the progenitors 2^{⋆4}:S_{4}, 2^{⋆4}:A_{4}, 2^{⋆3}:S_{3} and 2^{⋆3}:A_{3}, all third order relations in 2^{⋆3}:S_{3} and 2^{⋆3}:A_{3} and all relations of length up to 6 in 2^{⋆3}:A_{3}. In some progenitors, we have included some other relations which we felt were of some interest. We have excluded from consideration those parameters that can be obtained from other ones, for example, in the progenitor 2^{⋆5}:S_{5}, (0, 1, 2)(3, 4)t_{3} has order lcm(3, o((0, 1)t_{0})).

We note that some of the parameters must be even or divisible by 3. For example, in the progenitor 2^{⋆5}:A_{5}, the element (0, 1)(2, 3)t_{0} must have even order. Originally, we expressed this with the relation [(0, 1)(2, 3)t_{0}]^{d} = 1, but now we prefer [t_{0}t_{1}]^{d/2} = 1 instead. Similarly, in the progenitor 2^{⋆7}:L_{3}(2), we must have that (0, 1, 6)(2, 3, 5)t_{2} has order divisible 3, so we prefer to express the relation [(0, 1, 6)(2, 3, 5)t_{2}]^{b} = 1 as [t_{5}t_{3}t_{2}]^{b/3} = 1, or equivalently, as [t_{0}t_{1}t_{3}]^{b/3} = 1. Similar remarks apply to other relations of this type. We shall still regard such relations as being first order relations (or in the case of [t_{0}t_{2}t_{0}t_{6}]^{p/2} = 1 in 2^{⋆7}:L_{3}(2), a second order relation). For historical reasons, see [19], we have opted to keep the parameter e of 2^{⋆7}:L_{3}(2) even though the relations it expresses can be expressed using the d or the s parameters (if 4 | e then e = lcm(4, d), if 4 ∤ e then e = 2s and e can never be odd).

It is fairly easy to obtain presentations for our progenitors factored by our extra relations. For example, the progenitor 2^{⋆3}:S_{3} factored out by the relations given in Table 7.4 has the following presentation:

\begin{matrix}⟨x, y, t | x^{3} = & y^{2} = (yx)^{2} = 1 = t^{2} = [y, t] = (xt)^{a} \\ = & (yxt)^{b} = (yt^{x}t)^{c} = (xt^{xt})^{d} = (yxt^{xt})^{e}⟩. \end{matrix}

Extra relations for the progenitor 2^{⋆3}:A_{3}

Let G = ⟨x, t | x^{3} = t^{2} = 1⟩ ≅ 2^{⋆3}:A_{3} ≅ L_{2}(ℤ) and let q = xt and r = x^{2}t = x^{−1}t. Then the relations we factor out by up to the sixth order are the orders of the following 14 words of length up to 6 in q and r:

\begin{matrix}q (or r), qr, qqr, qqqr, qqrr, qqqqr, qqqrr, qqrqr, \\ qqqqqr, qqqqrr, qqqrrr, qqqrqr, qqrqrr and qqrrqr, \end{matrix}

which have orders a, b, c, d_{1}, d_{2}, e_{1}, e_{2}, e_{3}, f_{1}, f_{2}, f_{3}, f_{4}, f_{5} and f_{6} respectively. Applying the reciprocity automorphism α : (x, t) ↦ (x^{−1}, t) to the above words sends the first 12 words to words of the same order but sends qqrqrr (of order f_{5}) to a word (namely rrqrqq) of order f_{6} and vice-versa. We say that α fixes parameters a–f_{4} and interchanges f_{5} with f_{6}.

Naturally, if we factor out by a relation, say (xt)^{7} = 1, we may introduce many relations between the parameters. For example, when we factor out by a = 7 (so that we are considering images of the △-group (2, 3, 7)), we need only consider the 6 parameters a, b, d_{2}, e_{3}, f_{5} and f_{6}, for we have that a = d_{1} = 7, b = c = f_{2}, d_{2} = e_{2} = f_{4}, e_{1} = 3, e_{3} = f_{3} and f_{1} = 2.

7.3.5 The actual cases—generators of order 3 (permutation action)

We have not really considered all the progenitors in this section properly. However, we have included all first order relations for all the progenitors except 3^{⋆8}:L_{2}(7) and 3^{⋆8}:PGL_{2}(7) which we have not really studied at all, all second order relations for 3^{⋆3}:S_{3} and some s-type relations for 3^{⋆4}:S_{4}, 3^{⋆6}:L_{2}(5) and 3^{⋆7}:L_{3}(2). We have included other relations we have deemed to be useful and would like to consider others such as the order of t_{0}t_{1}^{−1} in 3^{⋆7}:L_{3}(2). These extra relations are shown in Table 7.5.

As an example of a presentation of a progenitor with its extra relations adjoined we take 3^{⋆4}:S_{4} factored out by the relations corresponding to parameters a–q in Table 7.5:

\begin{matrix}⟨x, y, t | x^{4} = & y^{2} = (xy)^{3} = 1 = t^{2} = [t, y] = [t^{x}, y] = (xt)^{a} = (yxt)^{b} \\ = & (x^{2}t)^{c} = (y^{x}t)^{d} = [t, x]^{e} = (yx[t, x])^{f} = (ytt^{x^{2}})^{p} = (y[t, x^{2}])^{q}⟩. \end{matrix}

To this presentation, adjoin the relation(s) y = (tt^{x})^{s_{1}} and y = [t, x]^{s_{2}} when the respective s-parameter(s) is/are in use.

7.3.6 The actual cases—generators of order 3 (monomial action)

For all the progenitors we are considering, we consider all their first order relations. For 3^{⋆3}:_{m} S_{3}, we consider all second order relations and one third order relation and for 3^{⋆2}:_{m} D_{8} we consider all second and third order relations. For all the other progenitors except 3^{⋆4}:_{m} S_{4}, we have kept to first order relations only. We have not used any s-type relations here, so any s’s that appear in the tables are ordinary n^{th} order parameters. All these relations (except a couple for 3^{⋆4}:_{m} S_{4}) are shown in Table 7.6.

For an example of a presentation of a progenitor factored out by its extra relations we consider 3^{⋆4}:_{m} S_{4} factored out by all the relations of Table 7.6 and the extra relations [(0, 1, 2)t_{0}t_{2}t_{1}]^{s} = 1 and [(2, ¯3)(0, ¯0)(1, ¯1)t_{0}t_{1}^{−1}]^{v} = 1.

\begin{matrix}⟨x, y, t | x^{4} = & y^{2} = (yx)^{3} = 1 = t^{3} = (yt)^{2} = (yt^{x})^{2} = (xt)^{a} \\ = & (xt^{2})^{b} = (y^{x}t)^{c} = (yxt)^{d} = (yxt^{2})^{e} = (x^{2}t)^{f} \\ = & (yxtt^{x})^{p} = (yt^{x^{2}}t)^{q} = (ytt^{x^{2}})^{r} = ((yxt)^{3}yx)^{s} = (ytt^{x})^{v}⟩. \end{matrix}

7.4 A few remarks

There are some facets of the results pertaining to these progenitors that we wish to examine in some detail, which we do in various sections below. In Section 7.5 we examine Coxeter presentations of some of the progenitors and in Section 7.6 we determine the

Table 7.4: Extra relations for the progenitors 2^{⋆n}:N

2^{⋆3}:S_{3}  2^{⋆4}:A_{4}  2^{⋆5}:A_{5}
[(0, 1, 2)t_{0}]^{a} = 1  [(0, 1, 2)t_{0}]^{a} = 1  [(0, 1, 2, 3, 4)t_{0}]^{a} = 1
[(0, 1)t_{0}]^{b} = 1  [(0, 1)(2, 3)t_{0}]^{b} = 1  [(0, 2, 4, 1, 3)t_{0}]^{b} = 1
[(0, 1)t_{0}t_{2}]^{c} = 1  [(0, 1, 2)t_{0}t_{3}]^{c} = 1  [(0, 1, 2)t_{0}]^{c} = 1
[(0, 1, 2)t_{0}t_{1}t_{0}]^{d} = 1  [(0, 1)(2, 3)t_{0}t_{2}]^{d} = 1  [t_{0}t_{1}]^{d/2} = 1
[(0, 1)t_{0}t_{2}t_{0}]^{e} = 1  [(0, 1)(2, 3)t_{0}t_{4}]^{e} = 1
2^{⋆4}:S_{4}  2^{⋆6}:L_{2}(5)  2^{⋆5}:S_{5}
[(0, 1, 2, 3)t_{0}]^{a} = 1  [(0, 1, 2, 3, 4)t_{0}]^{a} = 1  [(0, 1, 2)(3, 4)t_{0}]^{a} = 1
[(0, 1, 2)t_{0}]^{b} = 1  [(0, 2, 4, 1, 3)t_{0}]^{b} = 1  [(0, 1, 2, 3, 4)t_{0}]^{b} = 1
[(0, 1)(2, 3)t_{0}]^{c} = 1  [(∞, 0, 1)(2, 4, 3)t_{∞}]^{c} = 1  [(0, 1, 2, 3)t_{0}]^{c} = 1
[(0, 1)t_{0}]^{d} = 1  [(∞, 0, 1)(2, 4, 3)t_{2}]^{d} = 1  [(0, 1, 2)t_{0}]^{d} = 1
[(0, 1, 2, 3)t_{0}t_{1}]^{ϵ_{1}} = 1  [(∞, 0)(1, 4)t_{∞}]^{e} = 1  [(0, 1)t_{0}]^{e} = 1
[(0, 1, 2)t_{0}t_{3}]^{ϵ_{2}} = 1  [(∞, 0)(1, 4)t_{1}]^{f} = 1  [(0, 1, 2)t_{0}t_{3}]^{f} = 1
[(0, 1)t_{0}t_{2}]^{e} = 1  [t_{2}t_{3}]^{s} = (∞, 0)(1, 4)  [(0, 1)t_{0}t_{2}]^{p} = 1

[(0, 1, 2)t_{0}t_{1}t_{0}]^{f} = 1 [(0, 1)t_{0}t_{2}t_{0}]^{ϕ} = 1 [t_{0}t_{1}]^{s} = (2, 3)

2^{⋆7}:L_{3}(2)  2^{⋆8}:L_{2}(7)  2^{⋆8}:PGL_{2}(7)
[(0, 1, 2, 3, 4, 5, 6)t_{0}]^{a} = 1 [(0, 1, 2, 3, 4, 5, 6)t_{0}]^{a} = 1  [(∞, 2, 5, 4, 6, 1, 0, 3)t_{∞}]^{a} = 1
[t_{0}t_{1}t_{3}]^{b/3} = 1 [t_{∞}t_{0}]^{b/2} = 1 [(∞, 4, 0, 2, 6, 3, 5, 1)t_{∞}]^{b} = 1
[(0, 1, 6)(2, 3, 5)t_{0}]^{c} = 1 [(∞, 0, 6)(1, 3, 5)t_{∞}]^{c} = 1  [(0, 4, 1, 5, 2, 6, 3)t_{0}]^{c} = 1
[t_{0}t_{1}]^{d/2} = 1  [(∞, 6, 4, 2)(0, 3, 5, 1)t_{∞}]^{d} = 1 [(∞, 1, 5, 0, 2, 6)t_{∞}]^{d} = 1
[(2, 6)(4, 5)t_{1}t_{3}]^{e/2} = 1  [(∞, 5, 6, 0)(1, 3, 2, 4)t_{∞}]^{e} = 1
[(0, 3)(2, 6, 4, 5)t_{2}]^{f} = 1  [(∞, 5, 3)(0, 4, 1)t_{∞}]^{f} = 1
[t_{0}t_{2}t_{0}t_{6}]^{p/2} = 1  [t_{∞}t_{0}]^{p/2} = 1
[(2, 6)(4, 5)t_{1}t_{2}]^{q} = 1  [t_{∞}t_{0}]^{s} = (1, 6)(2, 5)(3, 4)

[t_{1}t_{3}]^{s} = (2, 6)(4, 5)

Table 7.5: Extra relations for the progenitors 3^{⋆n}:N (permutation action)

3^{⋆3}:S_{3}  3^{⋆4}:A_{4}  3^{⋆5}:A_{5}
[(0, 1, 2)t_{0}]^{a} = 1  [(0, 1, 2)t_{0}]^{a} = 1  [(0, 1, 2, 3, 4)t_{0}]^{a} = 1
[(0, 1)t_{0}]^{b} = 1  [(0, 2, 1)t_{0}]^{b} = 1  [(0, 2, 4, 1, 3)t_{0}]^{b} = 1
[t_{0}t_{1}^{−1}]^{c} = 1  [(0, 1)(2, 3)t_{0}]^{c} = 1 [(0, 1, 2)t_{0}]^{c} = 1
[(0, 1, 2)t_{0}t_{1}^{−1}]^{d} = 1  [t_{0}t_{1}]^{d/2} = 1
[(0, 1)t_{0}t_{2}]^{e} = 1  [t_{0}^{−1}t_{1}]^{e} = 1
[(0, 1)t_{0}t_{2}^{−1}]^{f} = 1  [(0, 1, 2)t_{0}^{−1}t_{1}]^{f} = 1
(4, 2, 1) = [t_{0}t_{3}]^{s_{1}}
(4, 2, 1) = [t_{0}^{−1}t_{3}]^{s_{2}}
3^{⋆4}:S_{4}  3^{⋆5}:S_{5}
[(0, 1, 2, 3)t_{0}]^{a} = 1  [(0, 1, 2)(3, 4)t_{0}]^{a} = 1
[(0, 1, 2)t_{0}]^{b} = 1  [(0, 1, 2, 3, 4)t_{0}]^{b} = 1
[(0, 1)(2, 3)t_{0}]^{c} = 1  [(0, 1, 2, 3)t_{0}]^{c} = 1
[(0, 1)t_{0}]^{d} = 1  [(0, 1, 2)t_{0}]^{d} = 1
[t_{0}^{−1}t_{1}]^{e} = 1  [(0, 1)t_{0}]^{e} = 1
[(0, 1, 2)t_{0}t_{1}^{−1}]^{f} = 1  [t_{0}^{−1}t_{1}]^{f} = 1
[(0, 1)t_{0}t_{2}]^{p} = 1  [(0, 1, 2)t_{0}^{−1}t_{1}]^{p} = 1
[(0, 1)t_{0}t_{2}^{−1}]^{q} = 1  [(0, 1)t_{0}t_{2}]^{q} = 1
[t_{0}t_{1}]^{s_{1}} = (2, 3)  [(0, 1)t_{0}t_{2}^{−1}]^{r} = 1

[t_{0}^{−1}t_{1}]^{s_{2}} = (2, 3)

3^{⋆6}:L_{2}(5)  3^{⋆7}:L_{3}(2)
[(0, 1, 2, 3, 4)t_{0}]^{a} = 1  [(0, 1, 2, 3, 4, 5, 6)t_{0}]^{a} = 1
[(0, 2, 4, 1, 3)t_{0}]^{b} = 1  [(0, 6, 5, 4, 3, 2, 1)t_{0}]^{b} = 1
[(∞, 0, 1)(2, 4, 3)t_{∞}]^{c} = 1 [t_{0}t_{1}t_{3}]^{c/3} = 1
[(∞, 0, 1)(2, 4, 3)t_{2}]^{d} = 1 [(0, 1, 6)(2, 3, 5)t_{0}]^{d} = 1
[(∞, 0)(1, 4)t_{∞}]^{e} = 1  [t_{0}t_{1}]^{e/2} = 1
[(∞, 0)(1, 4)t_{1}]^{f} = 1  [(2, 6)(4, 5)t_{1}t_{3}]^{f/2} = 1
[t_{∞}^{−1}t_{0}]^{p} = 1  [(0, 3)(2, 6, 4, 5)t_{2}]^{p} = 1
[t_{2}t_{3}]^{s_{1}} = (∞, 0)(1, 4)  [t_{1}t_{3}]^{s_{1}} = (2, 6)(4, 5)
[t_{2}t_{3}^{−1}]^{s_{2}} = (∞, 0)(1, 4)  [t_{1}t_{3}^{−1}]^{s_{2}} = (2, 6)(4, 5)

Table 7.6: Extra relations for the progenitors 3^{⋆n}:_{m} N (monomial action)

3^{⋆3}:_{m} D_{8}  3^{⋆3}:_{m} S_{3}  3^{⋆4}:_{m} S_{4}
[(0, 1, ¯0, ¯1)t_{0}]^{a} = 1  [(0, 1, 2)t_{0}]^{a} = 1  [(0, ¯1, 2, ¯3)t_{0}]^{a} = 1
[(0, 1)t_{0}]^{b} = 1  [(0, 1, 2)t_{0}^{−1}]^{b} = 1  [(0, ¯1, 2, ¯3)t_{0}^{−1}]^{b} = 1
[(1, ¯1)t_{0}t_{1}]^{c} = 1  [(0, ¯1)(2, ¯2)t_{0}]^{c} = 1  [(0, ¯1)(2, ¯2)(3, ¯3)t_{0}]^{c} = 1
[(0, 1)t_{0}t_{1}t_{0}^{−1}]^{d} = 1  [(0, 1, 2)t_{0}t_{1}^{−1}]^{d} = 1  [(0, 1, 2)t_{0}]^{d} = 1
[(0, 1, ¯0, ¯1)t_{0}t_{1}t_{0}]^{e} = 1 [t_{0}t_{1}]^{e} = 1  [(0, 1, 2)t_{0}^{−1}]^{e} = 1
[(0, ¯1)(2, ¯2)t_{0}t_{2}]^{f} = 1  [(0, 1)(2, 3)t_{0}]^{f} = 1
[(0, ¯1)(2, ¯2)t_{2}t_{0}]^{p} = 1  [(0, 1, 2)t_{0}t_{1}^{−1}]^{p} = 1
[(0, 1, 2)t_{0}t_{2}t_{1}]^{q} = 1  [(0, ¯1)(2, ¯2)(3, ¯3)t_{0}t_{2}]^{q} = 1

[(0, ¯1)(2, ¯2)(3, ¯3)t_{2}t_{0}]^{r} = 1

3^{⋆3}:_{m} S_{4}  3^{⋆4}:_{m} 2·S_{4}^{+}  3^{⋆6}:_{m} L_{2}(5)
[(0, 1, ¯0, ¯1)(2, ¯2)t_{0}]^{a} = 1 [(0, ¯1, ¯2, 3, ¯0, 1, 2, ¯3)t_{0}]^{a} = 1 [(0, 1, 2, 3, 4)t_{0}]^{a} = 1
[(0, 1, 2)t_{0}]^{b} = 1  [(0, ¯1, 2, ¯0, 1, ¯2)(3, ¯3)t_{0}]^{b} = 1 [(0, 1, 2, 3, 4)t_{0}^{−1}]^{b} = 1
[(1, ¯2)t_{1}]^{c} = 1  [(0, ¯2, ¯0, 2)(1, ¯3, ¯1, 3)t_{0}]^{c} = 1 [(0, 2, 4, 1, 3)t_{0}]^{c} = 1
[(0, 2, 1)t_{0}]^{d} = 1  [(0, 2, 4, 1, 3)t_{0}^{−1}]^{d} = 1
[(1, ¯1)(2, 3)t_{2}]^{e} = 1  [(∞, 0, 1)(2, 4, ¯3)t_{∞}]^{e} = 1

[(∞, 0, 1)(2, 4, ¯3)t_{∞}^{−1}]^{f} = 1 [(∞, 0, 1)(2, 4, ¯3)t_{2}]^{p} = 1 [(∞, 0, 1)(2, 4, ¯3)t_{2}^{−1}]^{q} = 1 [(∞, 0)(1, 4)(2, ¯2)(3, ¯3)t_{∞}]^{r} = 1 [(∞, 0)(1, 4)(2, ¯2)(3, ¯3)t_{1}]^{s} = 1

\begin{matrix}3^{⋆5}:_{m} S_{5} & 3^{⋆7}:_{m} L_{3}(2) \\ [(0, ¯1)(2, ¯3, 4, ¯2, 3, ¯4)t_{0}]^{a} = 1 & [(0, 1, 2, 3, 4, 5, 6)t_{0}]^{a} = 1 \\ [(0, ¯1)(2, ¯3, 4, ¯2, 3, ¯4)t_{0}^{−1}]^{b} = 1 & [(0, 6, 5, 4, 3, 2, 1)t_{0}]^{b} = 1 \\ [(0, ¯1)(2, ¯3, 4, ¯2, 3, ¯4)t_{2}]^{c} = 1 & [(1, ¯1)(3, ¯3)(2, ¯6)(4, 5)t_{2}]^{c} = 1 \\ [(0, 1, 2, 3, 4)t_{0}]^{d} = 1 & [(1, ¯1)(0, 3, ¯0, ¯3)(2, 6, 4, ¯5)t_{0}]^{d} = 1 \\ [(0, ¯1, 2, ¯3)(4, ¯4)t_{0}]^{e} = 1 & [(1, ¯1)(0, 3, ¯0, ¯3)(2, 6, 4, ¯5)t_{2}]^{e} = 1 \\ [(0, ¯1, 2, ¯3)(4, ¯4)t_{0}^{−1}]^{f} = 1 & [(1, ¯1)(0, 3, ¯0, ¯3)(2, 6, 4, ¯5)t_{2}^{−1}]^{f} = 1 \\ [(0, 1, 2)t_{0}]^{p} = 1 & [(0, ¯1, 6)(2, ¯3, ¯5)t_{0}]^{p} = 1 \\ [(0, 1)(2, 3)t_{0}]^{q} = 1 & [(0, ¯1, 6)(2, ¯3, ¯5)t_{0}^{−1}]^{q} = 1 \\ [(0, ¯1)(2, ¯2)(3, ¯3)(4, ¯4)t_{0}]^{r} = 1 & [(0, ¯1, 6)(2, ¯3, ¯5)t_{2}]^{r} = 1 \end{matrix}

outer automorphism group for each progenitor we are considering. We move on to consider which L_{2}(q) and PGL_{2}(q) are homomorphic images of our progenitors in Section 7.7 before finally giving tables of presentations obtained from our progenitors in Section E.1.

In this section, we consider more mundane matters such as useful subgroups (those that look like standard Coxeter groups), some general images of the progenitors, what improper images a progenitor has (not many) and the derived series of a progenitor (at least part of it).

7.4.1 Useful subgroups

There are two types of subgroup we would like to consider, namely subgroups whose order is known and so can be used for coset enumeration and subgroups useful for getting small (and faithful) permutation representations. We concentrate on the former. The obvious candidate for such a group is N. Also, it is useful to consider subgroups generated by subsets of the symmetric generators, the orders of these groups are often known beforehand. In the progenitors 3^{⋆n}:N, it is often useful to consider ⟨N, t_{0}t_{1}^{−1}⟩, but only for permutation representations since, in general, we would not expect to know enough about its presentation to determine its order directly and hence we would not be able to use coset enumeration over this subgroup to verify the order of G.

It is often useful to seek out subgroups satisfying various of the standard Coxeter presentations, for example G^{p,q,r} and (p, q, r; s). We give two examples of such (sub)groups. The first is the subgroup ⟨(0, 1, 2), t_{0}t_{1}t_{1}⟩ = ⟨x, t^{xt}⟩ of the progenitor 2^{⋆3}:S_{3} which can be shown to be a copy of (2, 3, d) (or an image thereof). For our second example we choose the subgroup ⟨t_{0}, t_{1}, (0, ¯1)(2, ¯2)(3, ¯3), (2, ¯3)(0, ¯0)(1, ¯1)⟩ of 3^{⋆4}:_{m} S_{4}. This subgroup is an image of 2^{⋆3}:S_{3} factored by the relations [(0, 1, 2)t_{0}]^{c} = [(0, 1)t_{1}]^{f} = [(0, 1)t_{0}t_{2}]^{v} = 1 and hence it is also an image of G^{3,c,f}.

7.4.2 General images/presentations

There are not many of these worth considering. An obvious way to obtain an image of a progenitor is to abelianise it; by that we mean abelianise the m^{⋆n} part of it, that is we adjoin a suitable number (one here) of relations [t_{i}, t_{j}] = 1. For example, 2^{⋆7}:L_{3}(2) has 2^{7}:L_{3}(2) ≅ 2 × 2^{3_{1}+3_{2}}:L_{3}(2) as an image which is obtained by adjoining (t_{0}t_{1})^{2} = 1 only. Also all progenitors 2^{⋆n}:N have S_{n+1} as an image obtained by setting t_{i} ∼ (⋆, i) and letting N fix ⋆ and act naturally on {0, …, n−1} and indeed we have already seen that 2^{⋆n}:S_{n} factored by [(0, 1)t_{0}]^{3} = 1 gives S_{n+1}.

From the Moore presentation, ⟨t_{0}, …, t_{n−1} | t_{i}^{3} = (t_{i}t_{j})^{2} = 1 (i ≠ j)⟩ ≅ A_{n+2}, we easily obtain that a permutation or quasi-monomial progenitor 3^{⋆n}:N factored by a suitable number of t_{i}t_{j}t_{i}t_{j} = 1 gives a group A_{n+2}:N. For example, we have:

\frac{3^{⋆8}:PGL_{2}(7)}{t_{0}t_{1}t_{0}t_{1} = 1} ≅ (A_{10} × L_{2}(7)):2 and \frac{3^{⋆8}:_{m} PGL_{2}(7)}{t_{0}t_{1}t_{0}t_{1} = 1} ≅ A_{10} × PGL_{2}(7),

and factoring by a further [(0, 1, 2, 3, 4, 5, 6)t_{0}]^{9} = 1 gives S_{10} and A_{10} × 2 respectively.

We also mention the following theorem that we proved in [3], but which we leave unproven here.

Theorem 7.4.1. Let n ⩾ 3 be an integer and let N act transitively on points, unordered pairs and unordered triples of distinct points. Then:

\frac{2^{⋆n}:N}{t_{0}t_{1}t_{2}t_{0}t_{1}t_{2} = 1} ≅ ℤ^{n−1}.(N × 2),

the extension being split if n is odd and non-split if n is even. Moreover, we have:

\frac{2^{⋆n}:N}{(t_{0}t_{1}t_{2})^{2} = (t_{0}t_{1})^{m} = 1} ≅ m^{n−1}.(N × 2),

the extension being split if m or n is odd and non-split if they are both even.

7.4.3 The derived series

It is essentially quite easy to calculate G/G′ for any finitely presented group G, since we may abelianise all the relations. We find that we are considering just two perfect progenitors, namely 3^{⋆6}:_{m} L_{2}(5) and 3^{⋆7}:_{m} L_{3}(2). However, we can do better than this and show that for each progenitor here with insoluble control subgroup we have that G′′ is perfect and the only such progenitor where G′ is not (necessarily) perfect is 3^{⋆5}:_{m} S_{5}. In the table below, for each progenitor G we are considering we give G/G′ and G/G′′.

Progenitor G/G′ G/G′′ Progenitor G/G′ G/G′′
2^{⋆2}:S_{2}  2^{2}  D_{2∞}  3^{⋆5}:A_{5}  3  3
2^{⋆3}:A_{3}  6  ℤ^{2}:6  3^{⋆6}:L_{2}(5)  3  3
2^{⋆3}:S_{3}  2^{2} S_{3} × S_{3}  3^{⋆5}:S_{5}  6  6
2^{⋆4}:A_{4}  6  2^{4}:6  3^{⋆7}:L_{3}(2)  3  3
2^{⋆4}:S_{4}  2^{2}  D_{12}  3^{⋆8}:L_{2}(7)  3  3
2^{⋆5}:A_{5}  2  2  3^{⋆8}:PGL_{2}(7) 6  6
2^{⋆6}:L_{2}(5)  2  2  3^{⋆2}:_{m} D_{8}  2^{2}  D_{8}
2^{⋆5}:S_{5}  2^{2}  2^{2}  3^{⋆3}:_{m} S_{3}  2 Dih(3^{2})
2^{⋆7}:L_{3}(2)  2  2  3^{⋆4}:_{m} S_{4}  2 Dih(3^{2})
2^{⋆8}:L_{2}(7)  2  2  3^{⋆3}:_{m} S_{4}  2  S_{3}
2^{⋆8}:PGL_{2}(7) 2^{2}  2^{2}  3^{⋆4}:_{m} 2·S_{4}^{+}  2  S_{3}
3^{⋆3}:S_{3}  6  3^{3}:6  3^{⋆6}:_{m} L_{2}(5)  1  1
3^{⋆4}:A_{4}  3^{2}  2^{6}:3^{2}  3^{⋆5}:_{m} S_{5}  2  S_{3}
3^{⋆4}:S_{4}  6  S_{3} × 3 3^{⋆7}:_{m} L_{3}(2)  1  1

For 2^{⋆3}:A_{3} we have that G′ = ⟨[x, t], [x^{−1}, t]⟩ ≅ ℤ ⋆ ℤ ≅ F_{2}, the free group on two generators, and for 2^{⋆2}:S_{2} we have G′′ = 1.

7.4.4 Improper images

Let P ≅ m^{⋆n}:N a progenitor we are considering and let ϕ : P → G be a homomorphism (from P) onto G. The all normal subgroups of N other than 1 either act transitively on {T_{0}, …, T_{n−1}} (since the corresponding permutation groups are primitive) or m = 3 and this normal subgroup fuses t_{i} and t_{i}^{−1} for some i. (The latter case occurs for example in 3^{⋆4}:_{m} 2·S_{4}^{+}.) So |G : N| ⩽ m unless Nϕ ≅ N. Let Nϕ ≅ N and suppose either that (T_{0}:N_{0})ϕ ≇ T_{0}:N_{0} or that |{T_{0}, …, T_{n−1}}| < n. If the former case we have that T_{0} ⩽ N and |G : N| = 1, and if the latter case we use the primitivity of N to show that T_{0} = T_{1} = ⋯ = T_{n−1} and so |G : N| ⩽ m. Therefore, we have shown that in all cases where we have an improper image we must have that N (strictly Nϕ) has index at most m in G.

7.5 Coxeter presentations of our progenitors

We restrict our attention to those progenitors that can be realised in a nice way as a Coxeter diagram group. Since such groups are generated by involutions, we must have that G/G′ is an elementary abelian 2-group for such progenitors G. This rules out all the progenitors 3^{⋆n}:N as well as 2^{⋆3}:A_{3} and 2^{⋆4}:A_{4}. For a nice presentation, we should also like N, N_{0} and T_{0}:N_{0} to be exhibited nicely, that is, each of these subgroups should also be generated by involutions. This excludes 2^{⋆5}:A_{5}, 2^{⋆8}:L_{2}(7) and 2^{⋆8}:PGL_{2}(7). Thus we are left with the progenitors 2^{⋆n}:S_{n} (all n), 2^{⋆6}:L_{2}(5), 2^{⋆7}:L_{3}(2) and the monomial progenitors 3^{⋆n}:_{m} N. However, for the monomial progenitors, it seems better to force N^{0} to be generated by involutions as well and then we are left with 3^{⋆2}:_{m} D_{8}, 3^{⋆3}:_{m} S_{3}, 3^{⋆3}:_{m} S_{4} and 3^{⋆4}:_{m} 2·S_{4}^{+} as possibilities in the monomial case.

The progenitors with particularly nice Coxeter presentations are the 2^{⋆n}:S_{n} and we give a typical presentation below:

⟨ [figure: a_{n−1}; a_{n−2}; a_{3}; a_{2}; a_{1}; b; ∞]⟩,

where a_{i} ∼ (i, i−1) for all i and b ∼ t_{0}. Two examples of Coxeter presentations of 2^{⋆n}:S_{n} with extra relations are:

⟨ [figure: a; b; c; r]| (abc)^{q} = 1⟩ ≅ \frac{2^{⋆3}:S_{3}}{[(0, 1, 2)t_{0}]^{q} = [(0, 1)t_{0}]^{r} = 1} ≅ G^{3,q,r},

which also illustrates the isomorphism with another type of Coxeter group and:

⟨ [figure: a; b; c; d; [8]]| a = (cd)^{4}⟩ ≅ \frac{2^{⋆4}:S_{4}}{(2, 3) = [t_{0}t_{1}]^{2}} ≅ PGL_{2}(7) ≅ L_{3}(2):2,

which appears in the Atlas [10] and is part of a sequence of L. H. Soicher presentations for groups in the Suzuki Chain. We give Coxeter presentations for the other progenitors (except 3^{⋆3}:_{m} S_{4}) that we have deemed have suitable such things in Table 7.7.

Table 7.7: Coxeter diagram presentations of some progenitors

2^{⋆6}:L_{2}(5)  ⟨ [figure: a; b; c; d; 5; ∞]| (abc)^{5} = 1⟩
a ∼ (1, 4)(2, 3), b ∼ (0, 1)(2, 4), c ∼ (∞, 0)(2, 3) and d ∼ t_{∞}

2^{⋆7}:L_{3}(2)  ⟨ [figure: a; b; c; d; e; [4]; ∞]| a = (ce)^{2}⟩

a ∼ (2, 5)(4, 6), b ∼ (1, 4)(3, 5), c ∼ (0, 1)(2, 5), d ∼ t_{0} and e ∼ (2, 4)(5, 6)

3^{⋆2}:D_{8}  ⟨ [figure: a; b; c; d; 4; ∞]| d = (ab)^{2}⟩
a ∼ (1, ¯1), b ∼ (0, 1), c ∼ −1.t_{0} and d ∼ −1 = (0, ¯0)(1, ¯1)

3^{⋆3}:_{m} S_{3}  [figure: a; b; c; ∞]
a ∼ (1, ¯2)(0, ¯0), b ∼ (0, ¯1)(2, ¯2) and c ∼ t_{0}

3^{⋆4}:_{m} 2·S_{4}^{+}  ⟨ [figure: a; b; c; d; e; [4]; ∞]| a = (ce)^{2}⟩

a ∼ −1, b ∼ −1.t_{0}, c ∼ (0, 1)(3, ¯3), d ∼ (1, 2)(3, ¯3) and e ∼ (1, ¯1)(2, 3)

7.6 Automorphisms of the progenitors

In this section, we consider the outer automorphism groups of the progenitors. In fact, we are only really interested in the more interesting outer automorphisms. That is, we will not be interested in inner automorphisms or those automorphisms that normalise N and the subset T̄ = {T_{0}, …, T_{n−1}}—these latter automorphisms extend N to N_{M(T,n)}(N). We shall only consider those automorphisms that swap N and T_{0}:N_{0} or that centralise N, normalise T_{0}:N_{0} but do not normalise T_{0}. We have seen in Sections 2.4 and 3.8 that these are essentially all the automorphisms there are. Moreover, the (slightly freakish) case when there is an automorphism swapping N and T_{0}:N_{0} can only occur when N ≅ T_{0}:N_{0}, so here just in the progenitor 3^{⋆3}:_{m} S_{3} (and also 2^{⋆2}:S_{2} ≅ 2 ⋆ 2 ≅ D_{2∞}).

There are more progenitors that possess a non-trivial automorphism centralising N, normalising T_{0}:N_{0} but not normalising T_{0}. (They are also more interesting.) Let α be such an automorphism and let s_{i} = t_{i}α and S_{i} = ⟨s_{i}⟩, S = {s_{0}, …, s_{n−1}} and S̄ = {S_{0}, …, S_{n−1}}. Then we may write our progenitor as (S_{0} ⋆ S_{1} ⋆ … ⋆ S_{n−1}):N ≅ S^{⋆n}:N (where S ≅ S_{i} ≅ T) and so the s_{i} could act as symmetric generators (which look much like the t_{i}). We refer to the s_{i} as auxiliary symmetric generators. The following progenitors have auxiliary symmetric generators:

2^{⋆3}:S_{3}, 3^{⋆4}:A_{4} and 3^{⋆4}:_{m} S_{4}.

The auxiliary symmetric generators of 2^{⋆3}:S_{3} are s_{i} = t_{i}(j, k) for {i, j, k} = {0, 1, 2} and the auxiliary symmetric generators of 3^{⋆4}:A_{4} and 3^{⋆4}:_{m} S_{4} are the elements t_{i}(j, k, l) where {i, j, k, l} = {0, 1, 2, 3}. In the latter two progenitors there are actually two classes of auxiliary symmetric generators corresponding to whether ijkl is an even permutation of 0123 or not.

In general, these exceptional automorphisms have a rather ‘wild’ effect on the parameters (they tend to fix very few or none whatsoever) compared to the rather ‘tame’ effect of elements of N_{M(T,n)}(N) which can fix many parameters, usually including most, if not all, first order parameters. Also it is usually easier to see that effect of elements of N_{M(T,n)}(N) on the parameters. We consider the progenitor P = 3^{⋆4}:_{m} S_{4} which has the automorphisms below:

α : t_{0} ↦ t_{0}(1, 2, 3), π ↦ π for all π ∈ N, and
β : t_{0} ↦ t_{0}^{−1}, π ↦ π for all π ∈ N,

where α is an exceptional automorphism and β ∈ N_{M(T,n)}(N). We visibly have that α^{3} = β^{2} = (αβ)^{2} = 1 and so we have ⟨α, β⟩ ≅ D_{6} and in fact we have Aut(P) = Inn(P):⟨α, β⟩ and so ⟨α, β⟩ contains all representative outer automorphisms. We easily determine that β permutes the parameters as (a, b)(c)(d, e)(f)(p)(q, r)(v)(s,?) and less easily determine that α permutes the parameters as (a, b, c)(d, f, e)(p,?,?)(q, r,?)(v,?,?)(s,?,?) where the question marks denote undefined parameters.

For each of the progenitors, P, with which we are dealing, we list Out(P) in the table below. Except for the progenitors mentioned above, we have that Out(P) ≅ N_{M(T,n)}(N)/N.

P  Out(P)  P  Out(P)  P  Out(P)
2^{⋆3}:A_{3}  2  2^{⋆8}:PGL_{2}(7)  1  3^{⋆8}:PGL_{2}(7)  2
2^{⋆3}:S_{3}  2  3^{⋆3}:S_{3}  2  3^{⋆2}:_{m} D_{8}  1
2^{⋆4}:A_{4}  2  3^{⋆4}:A_{4}  D_{12}  3^{⋆3}:_{m} S_{3}  2^{2}
2^{⋆4}:S_{4}  1  3^{⋆4}:S_{4}  2  3^{⋆4}:_{m} S_{4}  D_{6}
2^{⋆5}:A_{5}  2  3^{⋆5}:A_{5}  2^{2}  3^{⋆3}:_{m} S_{4}  2
2^{⋆6}:L_{2}(5)  2  3^{⋆6}:L_{2}(5)  2^{2}  3^{⋆4}:_{m} 2·S_{4}^{+}  1
2^{⋆5}:S_{5}  1  3^{⋆5}:S_{5}  2  3^{⋆6}:_{m} L_{2}(5)  4
2^{⋆7}:L_{3}(2)  1  3^{⋆7}:L_{3}(2)  2  3^{⋆5}:_{m} S_{5}  2
2^{⋆8}:L_{2}(7)  2  3^{⋆8}:L_{2}(7)  2^{2}  3^{⋆7}:_{m} L_{3}(2)  2

We remark that N_{M(T,n)}(N) can have a subtle structure in monomial progenitors, for example in 3^{⋆6}:_{m} L_{2}(5) we have that N_{M(T,n)}(N) ≅ (L_{2}(5) × 2)·2 ≅ (A_{5} × 2)·2 rather than PGL_{2}(5) × 2 ≅ S_{5} × 2.

7.7 L_{2}(q)’s and PGL_{2}(q)’s in our progenitors

Given a progenitor, P ≅ T^{⋆n}:N, it is natural to ask what finite images it has. In particular, which finite simple groups are images of P? Now ‘most’ finite simple groups are L_{2}(q) for some q. We therefore ask which L_{2}(q) are images of P. In fact, since it will not make the problem more complicated, we ask which L_{2}(q) and PGL_{2}(q) are images of P. Throughout this section, q = p^{m}, where p is prime (and m ∈ ℕ^{+}).

For the progenitors T^{⋆n}:N that we are studying, if G is a finite homomorphic image, then one of:

(i) G is generated by subgroups isomorphic to N and T_{0}:N_{0} intersecting in N_{0}.

(ii) G ≅ N is an image of 2^{⋆3}:S_{3}, 3^{⋆4}:A_{4} or 3^{⋆4}:_{m} S_{4} in which the images of N,

T_{0}:N_{0}, N_{0} and T_{0} are isomorphic to N, N_{0}, N_{0} and T_{0} respectively.
(iii) The images of the T_{i} are all equal.
or (iv) The images of the T_{i} are trivial and G is a homomorphic image of N.

Images that arise in ways (iii) and (iv) above are said to be trivial whereas those that arise in ways (i) and (ii) above are said to be non-trivial images. We shall only be interested in which L_{2}(q) and PGL_{2}(q) are non-trivial images of our progenitors.

Our method is roughly to consider all configurations (if any) of subgroups isomorphic to N and T_{0}:N_{0} intersecting in N_{0} inside PGL_{2}(q) (or L_{2}(q)) and determine which group they generate. To this end, we invoke Dickson’s Theorem on the conjugacy classes of subgroups of L_{2}(q) and PGL_{2}(q). The part of this we shall need is stated below.

Theorem 7.7.1. (Dickson [20]) Subgroups of PGL_{2}(q) other than L_{2}(q) and PGL_{2}(q) are subgroups of one of the following:

(i) The Frobenius group p^{m}:C_{q−1} ≅ E_{q}:C_{q−1}. (N.B., q = p^{m}.)
(ii) The dihedral groups D_{2(q+1)} and D_{2(q−1)}.
(iii) The groups A_{4} (tetrahedral), S_{4} (octahedral) and A_{5} (icosahedral).
(iv) The subfield subgroups L_{2}(r) and PGL_{2}(r) where q is a non-trivial power of r.

Furthermore, each of the subgroups mentioned is unique up to conjugacy in PGL_{2}(q).

When dealing with some progenitors, we needed the following the result, which is number theoretical in nature (in some sense).

Theorem 7.7.2. Let F = 𝔽_{q} be a finite field not in characteristic 2 and let κ ∈ F \ {0}. Then the equation:

α^{2} + β^{2} = κ,

has q − ε solutions for (α, β) in F × F where ε ∈ {±1} and q ≡ ε (mod 4).

Proof. Let n_{κ} denote the number of solutions of α^{2} + β^{2} = κ. If q ≡ 1 (mod 4) then α^{2} + β^{2} = (α + βi)(α − βi) where i = \sqrt{−1} ∈ F. Thus n_{κ} = (q − 1) for all κ ≠ 0 if q ≡ 1 (mod 4).

Now if q ≡ 3 (mod 4) we work in F̃ = F (i) = 𝔽_{q^{2}}. Then F̃ has a field automorphism, σ say, that fixes F = 𝔽_{q} and negates i. Thus σ : a ↦ a^{q} for all a ∈ F̃. For a ∈ F̃, let a = α + βi where α, β ∈ F. Then aa^{σ} = α^{2} + β^{2} is the norm of a, and for each non-zero κ ∈ F there are precisely q + 1 values of a such that a has norm κ. Thus the equation κ = α^{2} + β^{2} has q + 1 solutions. This gives the result when q ≡ 3 (mod 4). □

For some progenitors (the small ones), our aim is to show that ‘most’ L_{2}(q) and PGL_{2}(q) are images. When this is the case, it is quite often useful to try to construct a suitable matrix representation of the progenitor. It is often easier to make use of the isomorphism SO_{3}(q) ≅ PGL_{2}(q) and work a 3-dimensional orthogonal representation of the group rather than work projectively in a 2-dimensional representation. We now consider the individual cases.

7.7.1 The trivial cases

These are the progenitors in which T_{0}:N_{0} is never a subgroup of PGL_{2}(q). The only images L_{2}(q) and PGL_{2}(q) in these progenitors are the trivial ones. The cases are: 2^{⋆5}:A_{5}, 2^{⋆5}:S_{5}, 2^{⋆7}:L_{3}(2), 2^{⋆8}:L_{2}(7), 2^{⋆8}:PGL_{2}(7), 3^{⋆4}:S_{4}, 3^{⋆5}:A_{5}, 3^{⋆6}:L_{2}(5), 3^{⋆5}:S_{5}, 3^{⋆7}:L_{3}(2), 3^{⋆8}:L_{2}(7), 3^{⋆8}:PGL_{2}(7), 3^{⋆3}:_{m} S_{4} (both cases), 3^{⋆4}:_{m} 2·S_{4}^{+}, 3^{⋆5}:_{m} S_{5} and 3^{⋆7}:_{m} L_{3}(2).

7.7.2 The cases 3^{⋆4}:A_{4} and 3^{⋆4}:_{m} S_{4}

In these cases, T_{0}:N_{0} is a subgroup of PGL_{2}(q) only in characteristic 3. The (only) [PG]L_{2}(q)-images of 3^{⋆4}:A_{4} are L_{2}(3^{m}) and the only [PG]L_{2}(q)-images of 3^{⋆4}:_{m} S_{4} are PGL_{2}(3^{2m−1}) and L_{2}(9^{m}). (In all cases m ⩾ 1 is an integer.)

7.7.3 The cases 2^{⋆4}:A_{4} and 2^{⋆4}:S_{4}

In these cases N can be chosen unique up to conjugacy in PGL_{2}(q). Then N_{0} = N^{0} can be chosen uniquely within N. Then, except in characteristics 2 and 3, there is a unique involution, t, that commutes with N_{0}. From the uniqueness of the situation, we see that this configuration generates a subgroup of the subfield subgroup PGL_{2}(p).

The result is that 2^{⋆4}:A_{4} has [PG]L_{2}(q)-images L_{2}(p) (p ≡ ±1 mod 12) and PGL_{2}(p) (p ≡ ±5 mod 12) only.

The progenitor 2^{⋆4}:S_{4} has [PG]L_{2}(q)-images L_{2}(p) (p ≡ ±1 mod 24) and PGL_{2}(p) (p ≡ ±5, ±7, ±11 mod 24) only.

7.7.4 The cases 2^{⋆6}:L_{2}(5) and 3^{⋆6}:_{m} L_{2}(5)

The analysis here is much the same as the previous case. That is, the configuration of N ≅ A_{5} and m:D_{10} intersecting in D_{10} is unique whenever it exists. The only proper subgroups that can be generated by such configurations are the subfield subgroups, so the q that we seek are the smallest suitable q in each characteristic (except 5 and m). The progenitor 3^{⋆6}:_{m} L_{2}(5) is perfect, so we always get L_{2}(q) in this case, and in the 2^{⋆6}:L_{2}(5) case, we get L_{2}(q) if and only if we have D_{20} ⩽ L_{2}(q), otherwise we get PGL_{2}(q).

We have that 2^{⋆6}:L_{2}(5) has [PG]L_{2}(q)-images L_{2}(p) (p ≡ ±1 mod 20), PGL_{2}(p) (p ≡ ±9 mod 20) and PGL_{2}(p^{2}) (p ≡ ±3 mod 10).

The progenitor 3^{⋆6}:_{m} L_{2}(5) has [PG]L_{2}(q)-images L_{2}(16), L_{2}(p) (p ≡ ±1 mod 30), L_{2}(p^{2}) (p ≡ ±11 mod 30) and L_{2}(p^{4}) (p ≡ ±7, ±13 mod 30).

7.7.5 The case 3^{⋆3}:S_{3}

Somewhat surprisingly, the required configuration of N ≅ S_{3} and T_{0}:N_{0} ≅ 6 intersecting in N_{0} = N^{0} is unique. We rule out characteristics 2 and 3 because of the absence of elements of order 6 in PGL_{2}(q). In characteristics other than 2 and 3, PGL_{2}(q) has two classes of S_{3}, corresponding to the cases when S_{3} ⩽ L_{2}(q) and S_{3} ⩽̸ L_{2}(q), but in only one of those cases does an involution (in S_{3}) commute with an element of order 3. This element is unique up to inversion. Thus we have a unique situation where this can happen, and it happens in the subfield subgroup PGL_{2}(p).

The following are the only [PG]L_{2}(q) that are images of 3^{⋆3}:S_{3}: L_{2}(p) for p ≡ ±1 (mod 12) and PGL_{2}(p) for p ≡ ±5 (mod 12).

7.7.6 The case 3^{⋆2}:_{m} D_{8}

This time we seek subgroups isomorphic to D_{12} and D_{8} intersecting in 2^{2}, in which the central involution of the D_{12} is a non-central involution of D_{8} (since we are looking for groups with trivial centres). Thus we are not in characteristics 2 and 3. If q ≡ ±3 (mod 8) then PGL_{2}(q) has just one class of D_{8}, which is not is a subgroup of L_{2}(q), but if q ≡ ±1 (mod 8) then PGL_{2}(q) has two classes of D_{8}, one of which is a subgroup of L_{2}(q) and one of which is not. In the case q ≡ ±3 (mod 8), just two of the non-central involutions commute with an element of order 3 (and so can be central in a D_{12}) and both these involutions are conjugate in D_{8}. In the case q ≡ ±1 (mod 8), all elements of order 4 in PGL_{2}(q) are in the simple group L_{2}(q) and so just one of the conjugacy classes of D_{8} contains non-central involutions commuting with an element of order 3. In fact, such a D_{8} contains 4 such involutions, and they are all conjugate in N_{PGL_{2}(q)}(D_{8}) ≅ D_{16}.

Thus D_{12}, generated by this (essentially unique) non-central involution, v, of the D_{8}, an element, t, of order 3 commuting with this involution, and the central involution, c, of the D_{8} (which must invert t since c ∈ C_{PGL_{2}(q)}(v) which is dihedral), is unique. This D_{12} must occur in the subfield subgroup PGL_{2}(p). Thus the only [PG]L_{2}(q) which are images of 3^{⋆2}:_{m} D_{8} are: L_{2}(p) for p ≡ ±1 (mod 24) and PGL_{2}(p) for p ≡ ±5, ±7, ±11 (mod 24).

7.7.7 The case 2^{⋆3}:A_{3} ≅ 3^{⋆2}:S_{2} ≅ 3^{⋆2}:_{m} S_{2} ≅ 2 ⋆ 3 ≅ L_{2}(ℤ)

Let G ≅ L_{2}(q) or PGL_{2}(q). Then, for n ⩾ 4, n ≠ 6, ξ_{PGL_{2}(q)}(2X, 3A, nY) = 1 whenever classes 2X and nY are both in or both not in L_{2}(q). Choosing n = \frac{1}{2}(q ± 1) or q ± 1 guarantees that the (2X, 3A, nY)-triple generates G ≅ L_{2}(q) or PGL_{2}(q) (respectively) except in the very smallest cases when n ⩽ 6. We note that L_{2}(p) is (2, 3, p)-generated for all p. The result is that G ≅ [PG]L_{2}(q) is (2, 3)-generated except for the cases G ≅ PGL_{2}(5) ≅ S_{5} and G ≅ L_{2}(9) ≅ A_{6}.

7.7.8 The case 2^{⋆3}:S_{3} ≅ 3^{⋆2}:_{m} 2^{2} ≅ PGL_{2}(ℤ)

The question of which [PG]L_{2}(q) are images of 2^{⋆3}:S_{3} was studied in my M. Phil. and answered for the groups L_{2}(2^{m}) ≅ PGL_{2}(2^{m}) and L_{2}(p). The general result was left as a conjecture which we have now proved and we present the result below.

Theorem 7.7.3. Each of the groups L_{2}(q) and PGL_{2}(q) is a homomorphic image of the progenitor 2^{⋆3}:S_{3} except the following: L_{2}(3^{2m+1}), L_{2}(7), L_{2}(9) ≅ A_{6}, L_{2}(11) and PGL_{2}(5) ≅ S_{5}.

The exceptions are easy to deal with, for the groups L_{2}(3^{2m+1}) contain no S_{3}, and for the remaining four cases, we need only consider all possibilities for S_{3} and 2^{2} intersecting in 2 to check that they do not generate G (one of the groups L_{2}(7), L_{2}(9), L_{2}(11) and PGL_{2}(5)). The method of proof used in my M. Phil. was to consider all possible configurations of S_{3} and 2^{2} intersecting in 2, to see how many of these could possibly generate a proper subgroup and to deduce that some (in fact most) of them must generate G.

Our method of proof is to construct explicit matrix representations of the groups. The essential uniqueness of these constructions, which we shall sketch a proof of in Section 7.7.8.4, means that we can deal with the exceptions this way as well. For convenience, we have divided the proof into three cases, each of which is dealt with in a separate (subsub-)section below.

Throughout, x, y and t will satisfy the following presentation of 2^{⋆3}:S_{3}:

⟨x, y, t | x^{3} = y^{2} = (xy)^{2} = 1 = t^{2} = [t, y]⟩,

where ⟨x, y⟩ ≅ S_{3} and ⟨y, t⟩ ≅ 2^{2}.

7.7.8.1 The case L_{2}(q) ≅ PGL_{2}(q) for q even

Let F = 𝔽_{q} be a field in characteristic 2. The representation we construct is shown below:

x ∼ [ \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ] , y ∼ [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ] and t ∼ [ \begin{matrix}1 & 0 \\ a & 1 \end{matrix} ] ,

where a is not in any proper subfield. The presence of subgroups S_{3} and 2^{2} ensures that the only proper subgroup of G ≅ L_{2}(q) generated by x, y and t can be a subfield subgroup and the fact that xt has trace a+1 precludes this possibility. Thus ⟨x, y, t⟩ ≅ L_{2}(q) ≅ PGL_{2}(q).

7.7.8.2 The case PGL_{2}(q) for q odd

In this case, we construct the 3-dimensional orthogonal representation over F = 𝔽_{q} shown below:

x ∼ [ \begin{matrix}0 & 1 & 0 \\ −1 & −1 & 0 \\ a & a + 1 & 1 \end{matrix} ] , y ∼ [ \begin{matrix}1 & 0 & 0 \\ −1 & −1 & 0 \\ 1 & 0 & −1 \end{matrix} ] and t ∼ [ \begin{matrix}−1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ] ,

for some element a ∈ F. These matrices can be seen to preserve the (orthogonal) form:

B = [ \begin{matrix}2 & −1 & 1 \\ −1 & 2 & −a − 1 \\ 1 & −a − 1 & 2 \end{matrix} ] ,

which can be seen to have determinant −2(a + 2)(a − 1) which is non-zero except when a = −2 or 1. Now xt has characteristic polynomial (z − 1)(z^{2} − az + 1), and so if we let a = ζ + ζ^{−1}, where ζ is a primitive (q + 1)^{th} root of unity, then a ∈ F \ {−2, 1} (except if q = 5) and xt has order q + 1. This is enough to ensure that ⟨x, y, t⟩ ≅ PGL_{2}(q) except when q = 5.

7.7.8.3 The case L_{2}(q) for q odd

We use the same matrices as we used for the PGL_{2}(q) case, since these matrices are (essentially) unique. Now ⟨x, y, t⟩ contains subgroups isomorphic to S_{3} and 2^{2}, so it is never a subgroup of a Frobenius group, and if it is a subgroup of a dihedral group, then yt or t must be in the centre of that group and so ⟨x, y, t⟩ ≅ D_{12} in that case. Thus, if ⟨x, y, t⟩ is not [PG]L_{2}(q) or a subfield subgroup, we must have o(xt) ⩽ 6 and o(xyt) ⩽ 6. But xt has minimal polynomial (z − 1)(z^{2} − az + 1) and xyt has minimal polynomial (z − 1)(z^{2} − (−a−1) + 1), which allows us to tabulate the values of a for which this occurs below. (Actually, the table is not quite accurate when p = 3, for then we must replace p^{2}:D_{12} by 3:D_{12} ≅ S_{3} × S_{3}. It is not accurate when p = 2 either, but we have already dealt with that case and so we are ignoring it here.)

a  −2  1  −1 0 b5 b5^{∗} 2 −3
−a − 1  1  −2  0 −1 b5^{∗} b5 −3 2
o(xt)  2p  6  3 4 5 5 p  ?
o(xyt)  6  2p  4 3 5 5 ?  p
⟨x, y, t⟩ p^{2}:D_{12} p^{2}:D_{12} S_{4}  S_{4}  A_{5}  A_{5} see below

In the table, we included the cases a = 2 and −3, which is (more-or-less) what you obtain by reducing the ‘natural’ representation of 2^{⋆3}:S_{3} ≅ PGL_{2}(ℤ) modulo p and we get the groups L_{2}(p) for p ≡ 1 (mod 4) and PGL_{2}(p) for p ≡ 3 (mod 4). Now ⟨x, y, t⟩ is contained in a (proper) subfield subgroup if and only if a is in a proper subfield of F, since xt has trace a + 1. We must now determine when x, y and t are in L_{2}(q), which necessitates calculation of the spinor norm.

We consider the reflections:

r_{1} ∼ [ \begin{matrix}−1 & 0 & 0 \\ 1 & 1 & 0 \\ −1 & 0 & 1 \end{matrix} ] r_{2} ∼ [ \begin{matrix}1 & 0 & 0 \\ −1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ] and r_{3} ∼ [ \begin{matrix}1 & 0 & 0 \\ 0 & 0 & −1 \\ 0 & −1 & 0 \end{matrix} ] ,

which are reflections in the vectors v_{1} = (1, 0, 0), v_{2} = (1, 1, −1) and v_{3} = (0, 1, 1) respectively and these vectors have norms Q(v) = \frac{1}{2}β(v, v) of 1, a + 2 and 1 − a (respectively) when calculated with respect to B. These reflections, which are −y, −yt and −t respectively, can be seen to preserve B. Thus y = r_{1}r_{2}, t = r_{2}r_{3} and yt = r_{1}r_{3}, with spinor norms (modulo (F^{×})^{2}) of a + 2, (a + 2)(1 − a) and 1 − a respectively. Thus a necessary and sufficient condition that ⟨x, y, t⟩ ⩽ Ω_{3}(q) ≅ L_{2}(q) is that a + 2 and 1 − a are both squares in F.

We dispose of the characteristic 3 case first. In this case the requirement that these matrices lie in L_{2}(F) is that a + 2 and 1 − a = −(a + 2) are both squares in F. This is clearly impossible unless i = \sqrt{−1} ∈ F or a = −2, but the latter is one of the ‘disallowed’ cases when the form B is singular. Therefore, among finite fields 𝔽_{q}, this deals with the cases q = 3^{2m+1}. When F = 𝔽_{9}, a + 2 is a square in F just when a = 0, 1, 2, b5 or b5^{∗} (or −1 or −2), and in all those cases the given matrices generate proper subgroups. If F = 𝔽_{q} where q = 9^{m} ≠ 9, so that 𝔽_{9} = 𝔽_{3}(i) < F, then whenever a does not lie in proper subfield of F and a + 2 is a square in F then the given matrices must generate the whole of L_{2}(F). Now a + 2 is a square in F for \frac{1}{2}(q + 1) values of a, and of these, at most 9^{n}(\frac{1}{9} + \frac{1}{27} + \frac{1}{81} + ⋯) = q/6 lie in a proper subfield. Thus there are at least q/3 ⩾ 27 suitable values of a and L_{2}(9^{m}) (for m ⩾ 2) is a homomorphic image of 2^{⋆3}:S_{3} in many different ways.

If F has characteristic other than 2 or 3 then Theorem 7.7.2 shows that there are at least \frac{1}{4}(q − 1) values of a such that a + 2 and 1 − a are both squares. (There are q ± 1 pairs (α, β) such that α^{2} + β^{2} = (a + 2) + (1 − a) = 3, giving approximately q/4 values for α^{2} = a + 2 after taking into account the uncertainty in the sign of α and β.)

If q = p then we must avoid the six values of a which are a = 0, 1, −1, −2, b5 and b5^{∗}. Thus there are at least \frac{1}{4}(p − 1) − 6 = \frac{1}{4}(p − 25) admissible values of a, i.e., at least \frac{1}{4}(p − 25) values of a such that our given matrices generate L_{2}(p). This shows that L_{2}(p) is a homomorphic image of 2^{⋆3}:S_{3} whenever p > 23. For p = 5, 13, 17, 19 and 23, if we take a = 2, 2, 2, 3 and 6 respectively then our matrices generate L_{2}(p).

If q > p, then we must avoid a = 0, 1, −1, −2, b5 and b5^{∗} and any a lying in a proper subfield of F = 𝔽_{q}. This gives at most q(\frac{1}{p} + \frac{1}{p^{2}} + ⋯) + 2 = \frac{q}{p−1} + 2 values of a to avoid. If p ⩾ 7, this gives at least \frac{1}{4}(q − 1) − \frac{1}{6}q − 2 = \frac{1}{12}(q − 15) > 0 admissible values of a. This proves the theorem when q ⩾ 7. If p = 5 and q ⩾ 5^{3} = 125 then we have to avoid at most q(\frac{1}{25} + \frac{1}{125} + ⋯) = \frac{1}{20}q values of a, so we have at least \frac{1}{4}(q − 1) − \frac{1}{20}q = \frac{1}{20}(4q − 5) > 0 admissible values of a. When q = 5^{2} = 25 then we have to avoid just the five values of a in 𝔽_{5} giving at least \frac{1}{4}(q − 1) − 5 = 1 admissible values of a. This proves the theorem in all cases.

7.7.8.4 Proof of uniqueness

Firstly, we note that since G = ⟨x, y, t⟩ has derived subgroup of index 4 and quotient 2^{2} our 3 × 3 matrices representing x, y and t have determinant ±1 (with +1 for x). We may multiply y and t by −I if necessary to ensure that y and t (and x) have determinant 1, but without altering ⟨x, y, t, −I⟩/⟨−I⟩, so we need only consider representations in which x, y and t have determinant 1.

For notational convenience, we shall call the field of our representation F (= 𝔽_{q}), where F does not have characteristic 2, and we shall let V = F^{3}. Furthermore, we shall assume the easy-to-prove result that non-trivial representations of G in dimensions 1 and 2 can do no better than represent D_{12}. It is then easy to see that any reducible representation of dimension 3 can do not better than represent a subgroup of the affine group F^{2}:D_{12}. Thus we shall restrict our attention to irreducible representations.

Now y and t both have order 2 and determinant 1, so they have eigenvalues 1, −1 and −1. We choose v ≠ 0 such that v is fixed by y, so that y acts on V/⟨v⟩ as −I. Since t and y commute, they are simultaneously diagonalisable, and since t ≠ y, we must have that vt = −v. The irreducibility criterion means that v and vx are linearly independent. From (xy)^{2} = 1 we get v = vyxyx = vxyx and thus vx^{−1} = vxy = a_{1}v − vx for some a_{1} ∈ F (since y acts as −I on V/⟨v⟩). Thus v⟨x, y⟩ has dimension 2 and the equations x^{3} = (yx)^{2} = 1 force a_{1} = −1. Thus {v, vx, vxt} must be a basis for V. With respect to this basis we get:

x ∼ [ \begin{matrix}0 & 1 & 0 \\ −1 & −1 & 0 \\ a & b & 1 \end{matrix} ] , y ∼ [ \begin{matrix}1 & 0 & 0 \\ −1 & −1 & 0 \\ c & 0 & −1 \end{matrix} ] and t ∼ [ \begin{matrix}−1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ] ,

and the conditions that [t, y] = 1 = x^{3} = (yx)^{2} force b = a + 1 and c = 1.

In the characteristic 2 case, the restriction to ⟨x, y⟩ ≅ S_{3} must be as 1 ⊕ 2 since these representations are in different blocks. We choose as our seed (which we call v) a non-zero fixed vector y that lies in the 2-dimensional submodule of ⟨x, y⟩. Then it is fairly easy to show that the conditions y^{2} = [t, y] = 1 imply that vt = v and hence we may assume that {v, vx, vxt} is a basis of V. With respect to this basis, x, y and t must have the same matrices as in the odd characteristic case. However, it turns out that the representation is reducible, since the vector ṽ = (a + 1, 1, 1) is fixed.

7.7.8.5 Images U_{3}(q), L_{3}(q) of 2^{⋆3}:S_{3}

Let F be a field with primitive cube roots of unity (in any characteristic). Then SL_{3}(F) ≅ 3·L_{3}(F) ≅ L̸_{3}(F). So we wish to consider cases where an image of 2^{⋆3}:S_{3} occurs projectively in L_{3}(F), but does not occur in SL_{3}(F). In any case, the preimages in of S_{3} and 2^{2} in SL_{3}(F) are 3 × S_{3} and 2^{2} × 3 respectively. Let x̄, ȳ and t̄ be the preimages of x, y and t respectively. Then we may assume that ⟨x̄, ȳ⟩ ≅ S_{3} and that t̄ has order 2. This forces ȳ and t̄ to commute. Thus ⟨x̄, ȳ, t̄⟩ is also an image of 2^{⋆3}:S_{3}. All such images are known (see above) and do not include the groups SL_{3}(F) or SU_{3}(F). Thus we conclude:

Theorem 7.7.4. There are no images L_{3}(q) or U_{3}(q) of the progenitor 2^{⋆3}:S_{3}.

We note, however, that most L_{3}(q) and U_{3}(q) are (2, 3)-generated, and in such a way so that the reciprocity map is an automorphism, and this gives rise to many images of 2^{⋆3}:S_{3} of the form L_{3}(q):2 and U_{3}(q):2.

7.7.9 The case 3^{⋆3}:_{m} S_{3}

As with the progenitor 2^{⋆3}:S_{3}, our aim here is to show that ‘most’ L_{2}(q) and PGL_{2}(q) are images (of 3^{⋆3}:_{m} S_{3}). In fact, unlike the 2^{⋆3}:S_{3} case, ‘most’ L_{3}(q) and U_{3}(q) are also images of 3^{⋆3}:_{m} S_{3}. The proof, as in the 2^{⋆3}:S_{3} case, will be by construction of matrix generators for the group in question which satisfy the relations of 3^{⋆3}:_{m} S_{3}. In fact, except possibly in characteristic 3, the given matrix generators will be (essentially) unique. As usual, F (perhaps equal to 𝔽_{q}) is the field over which we are writing our representation.

7.7.9.1 Dimension 2

We choose as our ‘seed’ vector any non-zero fixed vector of y and write our matrices with respect to the basis {v, vx}. We obtain:

x ∼ [ \begin{matrix}0 & 1 \\ −1 & −1 \end{matrix} ] , y ∼ [ \begin{matrix}1 & 0 \\ −1 & −1 \end{matrix} ] and t ∼ [ \begin{matrix}a & 2a + 1 \\ (−a^{2} − a − 1)/(2a + 1) & −a − 1 \end{matrix} ] ,

where a is any element of F. Reducing modulo 2, we get:

x ∼ [ \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ] , y ∼ [ \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ] and t ∼ [ \begin{matrix}a & 1 \\ a^{2} + a + 1 & a + 1 \end{matrix} ] .

We note that there are projective 2-dimensional representations of 3^{⋆3}:_{m} S_{3} that cannot be realised as 2 × 2 matrices (but not in characteristic 2).

7.7.9.2 Dimension 3

As in the 2^{⋆3}:S_{3} case, we may assume that all matrices in our representation of 3^{⋆3}:_{m} S_{3} have determinant 1. Except in characteristic 3, the restriction of an irreducible representation of 3^{⋆3}:_{m} S_{3} to both ⟨x, y⟩ ≅ S_{3} and ⟨y, t⟩ ≅ S_{3} is as 2 ⊕ 1^{−}. In an irreducible representation, the 2-dimensional submodules of ⟨x, y⟩ and ⟨y, t⟩ must intersect in a 1space, which must be the fixed space of y. We take our ‘seed’ vector v to be any non-zero vector in this 1-space. Spinning up with respect to the basis {v, vx, vt}, we obtain:

x ∼ [ \begin{matrix}0 & 1 & 0 \\ −1 & −1 & 0 \\ a & a − 1 & 1 \end{matrix} ] , y ∼ [ \begin{matrix}1 & 0 & 0 \\ −1 & −1 & 0 \\ −1 & 0 & −1 \end{matrix} ] and t ∼ [ \begin{matrix}0 & 0 & 1 \\ b & 1 & b − 1 \\ −1 & 0 & −1 \end{matrix} ] ,

where a and b are any elements of F.

This representation is irreducible just when −2ab + a + b + 4 ≠ 0. The representation fixes a non-singular symmetric bilinear (orthogonal) form just when a = b and −2(a − 2)(a + 1) ≠ 0. The matrix of the form is shown below:

B = [ \begin{matrix}2 & −1 & −1 \\ −1 & 2 & 1 − a \\ −1 & 1 − a & 2 \end{matrix} ] ,

which has determinant −2(a−2)(a+1). Let σ be an automorphism of F of order 2. Then the representation fixes a σ-Hermitian (σ-unitary) form just when b = a^{σ}. The form is:

B = [ \begin{matrix}2 & −1 & −1 \\ −1 & 2 & 1 − a^{σ} \\ −1 & 1 − a & 2 \end{matrix} ] ,

of determinant −2aa^{σ} + a + a^{σ} + 4, which we require to be non-zero for non-degeneracy of the form.

All projective 3-dimensional representations of 3^{⋆3}:_{m} S_{3} can be realised absolutely, i.e., as 3 × 3 matrices.

7.7.9.3 Some images

Theorem 7.7.5. All the groups L_{2}(q) and PGL_{2}(q) are homomorphic images of 3^{⋆3}:_{m} S_{3} except PGL_{2}(5), L_{2}(3^{2m+1}) and PGL_{2}(9^{m}).

Proof. We attend to the exceptions first. For the characteristic 3 cases, PGL_{2}(q) has just one class of S_{3}, which is a subgroup of L_{2}(q) if q = 9^{m}, but is not a subgroup of L_{2}(q) if q = 3^{2m+1}. The case of PGL_{2}(5) is easy to handle making use of the isomorphism PGL_{2}(5) ≅ S_{5}.

For the characteristic 2 case, we use the 2-dimensional construction of Section 7.7.9.1 where a is not in any proper subfield. The elements xt, xt^{−1} and xyt have traces a^{2} +1, a^{2} and a^{2} + a respectively. This is sufficient to show that these matrices generate the whole of L_{2}(F).

In the odd characteristic case, we make use of the orthogonal representation constructed in Section 7.7.9.2. (This is the case when a = b.) In the reducible cases when a = 2 or −1, these matrices generate ℤ^{2}:S_{3} in characteristic zero or p^{2}:S_{3} in characteristic p ≠ 3 or 3^{2}:2 in characteristic 3. Now [x, t] = 1 just when a = −1, so in the irreducible case, when we must generate a subgroup of SO_{3}(F) ≅ PGL_{2}(F) (F = 𝔽_{q}), our given matrices cannot generate a subgroup of a Frobenius group E_{q}:C_{q−1}, a dihedral group or A_{4}. Thus we are left with the cases that our matrices generate S_{4}, A_{5} or a subfield subgroup (or L_{2}(q) or PGL_{2}(q)).

Now S_{4} ≅ PGL_{2}(3) is generated just when a = 0 or 1 and A_{5} ≅ L_{2}(5) is generated just when a = −b5 or −b5^{∗}. (There are essentially unique ways of generating each of these groups, the former requires xt, xt^{−1} and xyt to have orders 2, 3 and 4 or 3, 2 and 4 respectively and the latter case requires xt, xt^{−1} and xyt to have orders 5, 5 and 3 respectively.) Furthermore, since xt, xt^{−1} and xyt have traces a^{2} − 1, a^{2} − 2a and −a^{2} + a + 1 respectively, the matrices are in a proper subfield subgroup just when a is in a proper subfield.

We have now dealt with the images PGL_{2}(3) and L_{2}(5), so we may assume that q ⩾ 7. Moreover, we may suppose that a ≠ 0, 1, −1, 2, −b5 or −b5^{∗} and that 𝔽_{p}(a) = 𝔽_{q} = F so that our matrices generate L_{2}(q) or PGL_{2}(q). Also, the matrices generate L_{2}(q) just when y ∈ L_{2}(q). Now y is the product of (B-orthogonal) reflections r_{1} and r_{2} where

r_{1} ∼ [ \begin{matrix}1 & 0 & 0 \\ −1 & 0 & −1 \\ −1 & −1 & 0 \end{matrix} ] and r_{2} ∼ [ \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ] ,

are reflections in (1, 1, 1) and (0, 1, −1) respectively. These vectors have norms 2 − a and 1 + a respectively, giving y a spinor norm of (2 − a)(1 + a) = −a^{2} + a + 2 = −[(a − \frac{1}{2})^{2} − \frac{9}{4}] modulo squares of F.

In characteristic 3, the spinor norm becomes −(a − \frac{1}{2})^{2} = −(a + 1)^{2}, which, since a ≠ −1, is a square in F just when i = \sqrt{−1} ∈ F which in the finite case is just when F = 𝔽_{q} for q = 9^{m}. This deals with the characteristic 3 case. From now on we shall assume that F does not have characteristic 2 or 3 (and that |F | ⩾ 7).

If the characteristic is greater than 3, the Theorem 7.7.2 implies that −[(a− \frac{1}{2})^{2} − \frac{9}{4}] = \frac{9}{4} − (a − \frac{1}{2})^{2} is a square for \frac{1}{2}(q + 1) and \frac{1}{2}(q + 3) values of a in the cases q ≡ 1 (mod 4) and q ≡ 3 (mod 4) respectively. (We can give an exact number since \frac{9}{4} = (\frac{3}{2})^{2} is always a square in F.) Thus there are \frac{1}{2}(q − 1) and \frac{1}{2}(q − 3) values of a such that \frac{9}{4} − (a − \frac{1}{2})^{2} is not a square in F in the cases q ≡ 1 (mod 4) and q ≡ 3 (mod 4) respectively. For the ‘disallowed’ values of a, namely 0, 1, −1, 2, −b5 and −b5^{∗}, the four values −1, 2, −b5 and −b5^{∗} always force \frac{9}{4} − (a − \frac{1}{2})^{2} to be a square, whereas for a = 0 or 1, \frac{9}{4} − (a − \frac{1}{2})^{2} may or may not be q square.

If q = p, this gives at least \frac{1}{2}(q + 1) − 6 admissible values of a when trying to generate L_{2}(p), so we have shown that this is possible when p ⩾ 13. For p = 7 and 11 we take a = 4 and 5 respectively. When trying to generate PGL_{2}(p) there are at least \frac{1}{2}(q − 3) − 2 admissible values of a, so we have shown that we can generate PGL_{2}(p) this way for p ⩾ 11. For p = 7 we can take a = 3 or 5.

If q > p, we are not allowed any a in a proper subfield. This accounts for at most q(\frac{1}{p} + \frac{1}{p^{2}} + ⋯) = \frac{q}{p−1} ⩽ \frac{1}{4}q values of a. Furthermore, we may still have to consider the disallowed values of −b5 and −b5^{∗} of a. Thus, when generating L_{2}(q) there are at least \frac{1}{2}(q + 1) − 2 − \frac{1}{4}q = \frac{1}{4}(q − 6) admissible values of a, but q ⩾ 25 so we can always find an a such that our matrices generate L_{2}(q). When generating PGL_{2}(q) there are at least \frac{1}{2}(q − 3) − \frac{1}{4}q = \frac{1}{4}(q − 6) > 0 admissible values of a. This completes the proof of the theorem. □

The following conjecture concerns which of the groups L_{3}(q) and U_{3}(q) are images of 3^{⋆3}:_{m} S_{3}. A proof of this conjecture should not be too hard given a list of maximal subgroups of L_{3}(q) and U_{3}(q) (something like the irreducible non-orthogonal subgroups of L_{3}(q) are subfield subgroups L_{3}(r), U_{3}(r), the ‘base’ stabiliser and some of its subgroups, some oddments such as L_{3}(2), A_{6}, 3^{2}:2 ≅ U_{3}(2)′ and A_{7} and the Singer cycle subgroups).

Conjecture 7.7.6. All the groups L_{3}(q) and U_{3}(q) are homomorphic images of 3^{⋆3}:_{m} S_{3} except L_{3}(4), U_{3}(2), U_{3}(3) and U_{3}(5).

Remark. Very few of the progenitors studied here have images L_{3}(q) or U_{3}(q) owing to the fact that either the progenitor does not possess a true 3-dimensional representation, or that any 3-dimensional representations a progenitor has are orthogonal. By a true representation, we mean one in the resulting matrix group is a true image of the progenitor concerned, see Definition 3.1.3.

7.8 Refinements of previous work

In my M. Phil. [3], some groups (particularly soluble ones) were not identified. Four of these cases were identified by Derek Holt (personal communication) and these are listed below. (There are, of course, many cases still unresolved, in particular, those images in which N ∩ ⟨T ⟩ = 1 as well as soluble images of ‘small’ progenitors.)

7.8.1 The case G ≅ [2^{9}]:L_{3}(2)

This group, featured in Table 3.7 on Page 103 of [3], was the result of quotienting out the progenitor 2^{⋆7}:L_{3}(2) by the relation [(0, 1, 2, 3, 4, 5, 6)t_{0}]^{7} = 1. This gave rise to a group H:L_{3}(2) where H was a group of order 2^{9} = 512.

We find that 1 < H′ < Z(H) < H < G are the only normal subgroups of G, where H′ ≅ 2^{3} and Z(H) ≅ 4^{3}. We also easily ascertain the isomorphisms G/Z(H) ≅ 2^{3}:L_{3}(2) and G/H′ ≅ 2^{3_{1}+3_{2}} :L_{3}(2) where, in the latter case, the 2^{3_{1}+3_{2}} is a uniserial module for the L_{3}(2) with two non-isomorphic composition factors of dimension 3. Finally, since Z(H) ≅ 4^{3}, it must be the case that Z(H)/H′ and H′ are isomorphic as modules for the complementary L_{3}(2). Thus G ≅ 2^{3_{1}}.2^{3_{1}+3_{2}} :L_{3}(2).

7.8.2 The case G ≅ [2^{18}]:3·S_{7}

This group, featured in Table 3.7 on Page 103 of [3], was the result of quotienting out the progenitor 2^{⋆7}:L_{3}(2) by the relations [(0, 1, 2, 3, 4, 5, 6)t_{0}]^{12} = [(0, 1, 6)(2, 3, 5)t_{0}]^{10} = [(0, 3)(2, 6, 4, 5)t_{0}]^{6} = 1. In [3], we showed that G ≅ [2^{18}]:3·S_{7} has 2^{6}:S_{7} as a homomorphic image, but made no further headway.

Now if the 3 of the 3·S_{7} acts trivially on the [2^{18}] then quotienting out by q = 7 (that is [(2, 6)(4, 5)t_{1}t_{2}]^{7} = 1) should yield [2^{18}]:S_{7}. But in fact we get 2^{6}:S_{7}. So G ≅ [2^{12}].2^{6}:3·S_{7} where the 3·S_{7} acts faithfully on the [2^{12}]. But the smallest faithful 𝔽_{2}-module for 3·S_{7} has dimension 12 (see the ‘modular Atlas’ [27, page 13]), so this [2^{12}] is elementary abelian. So we have [2^{18}] ≅ 2^{12}.2^{6}. The 3 of the 3·S_{7} centralises the quotient 2^{6}, but has trivial centraliser in the 2^{12}, so that C_{[2^{18}]}(3) is a complementary 2^{6} and also N_{G}(3) ≅ 2^{6}:3·S_{7}. But the Frattini subgroup Φ([2^{18}]) is centralised by the 3 and is a normal subgroup of G. So Φ([2^{18}]) = 1 or 2^{6} and in any case the 2^{6} is normalised by the 2^{12} and hence by G. Thus G ≅ 2^{6⊕12}:3·S_{7} ≅ (2^{6} × 2^{12}):3·S_{7}.

7.8.3 The case G ≅ [2^{10}3]:S_{5}

This group, featured in Table 4.3 on Page 130 of [3], was the result of quotienting out the progenitor 3^{⋆5}:S_{5} (permutation action) by the relation [(0, 1, 2)t_{0}]^{6} = 1.

In this group, we find that H, the normal closure in G of [(0, 1, 2)t_{0}]^{3}, is a special group of order 2^{10} = 1024 with Z(H) = H′ ≅ 2^{2}. We find that C_{G}(t) ≅ 3 × S_{5} is a complement to H in G. (This complement does not contain a copy of the control subgroup N ≅ S_{5}.) We find that both H′ and H/H′ are irreducible as 𝔽_{2}-modules for 3 × S_{5}, though neither module can be absolutely irreducible. Thus we find that G ≅ 2^{2|8}:(3 × S_{5}).

7.8.4 The case G ≅ 2.(2^{8}:L_{2}(17))

This group, featured in Table 5.1 on Page 146 of [3], was the result of quotienting out the progenitor 3^{⋆3}:_{m} S_{3} (monomial action) by the relations [(0, 1, 2)t_{0}]^{17} = [(0, 1, 2)t_{0}^{−1}]^{9} = [(0, 1)t_{0}]^{16} = [(0, 1)t_{0}t_{2}]^{3} = 1. In this group, I found a subgroup J = ⟨xtyxt, yx⟩ ≅ 2.L_{2}(17) of index 2^{8} = 256 and the permutation action of G over J established that G has 2^{8}:L_{2}(17) as a homomorphic image.

In fact, we find that K = ⟨xtyxt, (xtyxt)^{yx}⟩ ≅ L_{2}(17) is a subgroup of index 2 in J (and so J ≅ L_{2}(17)×2). Taking the permutation action of G over K (which is faithful), we easily establish the existence of a normal elementary abelian subgroup of order 2^{9}. Since G is perfect, this establishes the shape G ≅ 2^{1+8}:L_{2}(17) where the L_{2}(17) acts uniserially on the 2^{1+8} with a single submodule of dimension 1.

7.9 Tabulated results

This is what the rest of this chapter has been leading up to. However, to avoid having 20 consecutive pages of nothing but tables in the main of the text, I have placed the tables, and other lists of results, in Appendix E. However, they properly belong here, in the main body of the thesis. An explanation of the notation used in the tables, as well as proofs of selected results from the tables is also to be found in Appendix E. The tables are more-or-less the same as those in my M. Phil. [3], and include the refinements of the previous section.

A presentation of the McLaughlin group

8.1 Introduction

In this chapter, we aim to obtain a presentation of the McLaughlin group, McL, as a homomorphic image of a progenitor of the form 3^{⋆110}:_{m} M_{11}. As Jabbar tried to find a nice presentation for this group, some of this chapter duplicates portions of Jabbar’s thesis [26], in particular, the presentation of M_{11} is his, and the presentation of 3_{2}·U_{4}(3) here is related to his presentation of U_{4}(3). Also, the presentation of L_{4}(3):2_{2} is found in [26].

For this presentation, we should perhaps seek a ‘better’ presentation of the control subgroup N ≅ M_{11}. This presentation will play a central rôle when we come to consider the Lyons group Ly as a homomorphic image of 3^{⋆2520}:_{m} 2·A_{11} in Chapter 10. Also, looking at the ‘bigger’ group makes some of the phenomena observed in this chapter much clearer. However, time constraints have meant that we have been unable to find a ‘better’ presentation of M_{11} or to do the Lyons group presentation full justice. These will be high on our list of priorities when this thesis is completed.

8.2 The control subgroup and progenitor

Firstly, we must find a suitable presentation for the (proposed) control subgroup, M_{11}. Such a presentation occurs in Jabbar’s thesis [26, page 126]. Specifically, we have:

N = ⟨x, y | x^{8} = y^{5} = y^{x^{2}}y^{−2} = 1 = (xy)^{6} = (x^{3}y)^{11}⟩ ≅ M_{11},

where x ∼ (0, 1)(2, 3, 6, 9, X, 4, 7, 5) and y ∼ (1, 3, 9, 5, 4)(2, 6, 7, X, 8). Without the relator (x^{3}y)^{11} we obtain a presentation for M_{11} × 2.

Throughout this chapter, we write N_{i_{1}...i_{m}} for the (pointwise-)stabiliser of {i_{1}, …, i_{m}} in N in its 11 point action. We found that ⟨x^{2}, xyxyxy^{2}xy⟩ = N_{01} ≅ M_{9} and that ⟨x^{2}y^{−1}, x^{−1}y^{2}xyxy^{−2}x⟩ = N_{23}. We also found that ⟨x^{2}, y, xyxyxy^{2}xy⟩ = ⟨x^{2}, xyxyxy^{2}x⟩ = N_{0} ≅ M_{10}. We shall also let S be the Steiner System S(4, 5, 11) that is preserved by N.

This Steiner System is compatible with Curtis’s ‘Kitten’ [12] construction of S(5, 6, 12) upon stabilising ∞.

The permutation action of N we wish to consider is its action on ordered pairs from {0, 1, …, X} and so we write t_{ij} and t_{ij}^{−1} for the symmetric generators corresponding to the ordered pair (i, j). However, the progenitor 3^{⋆110}:M_{11} we wish to construct is a monomial one and so we must specify the action of N_{01} on ⟨t_{01}⟩.

The monomial action N on the symmetric generators is specified by requiring that:

⟨t_{01}, N_{0}⟩

is the quasi-monomial form of 3^{⋆10}:M_{10} in which the outer elements of M_{10} permute and invert the symmetric generators. This forces

⟨t_{10}, N_{0}⟩

to be the variety of 3^{⋆10}:M_{10} in which M_{10} has a true-monomial action on the symmetric generators.

The monomial action of M_{11} on the symmetric generators is thoroughly horrible and we avoid it as much as possible. Therefore, when we obtain presentations for subgroups, we ensure that the labelling of a symmetric generator and its inverse is not important. Also, the two classes of elements of order 8 in M_{11} are thoroughly distinct (though they are automorphic in M_{10}). We have ensured that such distinctions do not matter.

8.3 Presentations of some subgroups of McL

In this section we find presentations for 3_{2}·U_{4}(3), 3^{4}:M_{10} and 5{1+2 \atop +}:3 which are subgroups of one or both of McL and 3·McL. We also consider some related presentations. In particular, we obtain presentations for L_{4}(3):2_{2} and L_{3}(4):2_{1} as images of 3^{⋆10}:M_{10} and 2^{⋆10}:M_{10} respectively.

8.3.1 A presentation of 3_{2}·U_{4}(3) and related presentations

We found that:

G = \frac{3^{⋆10}:_{m} M_{10}}{[(1, 2, 3, 6, 5, 4, 7, X)(8, 9)t_{1}]^{5} = 1} ≅ 3_{2}·U_{4}(3),

where the 3^{⋆10}:_{m} M_{10} is the one in which outer elements of M_{10} permute and invert the symmetric generators, the inner elements merely permuting them. A presentation of G is found to be:

⟨x, y, t | x^{8} = y^{4} = (xy)^{3} = 1 = (x^{−3}y)^{2} = (x^{4}y^{2})^{5} = t^{3} = [t, xy^{−1}] = t^{x^{2}y}t = (tx)^{5}⟩,

where x ∼ (1, 2, 3, 6, 5, 4, 7, X)(8, 9), y ∼ (1, 2, X, 3)(6, 9, 8, 7) and t ∼ t_{1}. A faithful permutation representation of G may be obtained over ⟨t_{1}, t_{2}, t_{3}, t_{9}⟩ ≅ U_{3}(3). (All triples and all special tetrads of symmetric generators generate copies of U_{3}(3). All non-special tetrads of symmetric generators generate the whole of G.) For i ∈ {1, 3, 5, 7}, we found that x^{i}t and x^{i}t^{−1} had order 5 and also that replacing (xt)^{5} = 1 by any of the 8 relations (x^{i}t^{±1})^{5} = 1 still yielded a presentation for G. This will be the subgroup ⟨t_{01}, N_{0}⟩ of McL. This case is dealt with in some detail in Chapter 5.

Incidentally, if we take the permutation version of 3^{⋆10}:M_{10}, we get:

G = \frac{3^{⋆10}:M_{10}}{[(1, 2, 3, 6, 5, 4, 7, X)(8, 9)t_{8}]^{4} = 1} ≅ L_{4}(3):2_{2}.

The relation we append here is also satisfied in the 3_{2}·U_{4}(3) case but does not seem to be sufficient to define a presentation in that case. In this case, special tetrads of symmetric generators generate copies of L_{3}(3) and non-special tetrads generate the whole of G.

Also mentioned in Chapter 5 is the presentation:

G = \frac{2^{⋆10}:M_{10}}{[(1, 2, 3, 6, 5, 4, 7, X)(8, 9)t_{1}]^{5} = 1} ≅ L_{3}(4):2_{1}.

Here, triples and special tetrads of symmetric generators generate PGL_{2}(7) and nonspecial tetrads generate the whole of G. A slightly weaker version is:

G = \frac{2^{⋆10}:M_{10}}{[(1, 2, 3, 6, 5, 4, 7, X)(8, 9)t_{8}]^{4} = 1} ≅ 2·L_{3}(4).2^{2}.

Triples and special tetrads of symmetric generators generate PGL_{2}(7), non-special tetrads generate 2·L_{3}(4):2_{1}. This group has a subgroup H = 2·L_{3}(4):2_{1} of index 2 to which we append an element of order 4 which squares to the central element v of H, centralises H′ and multiplies outer elements of H by v.

We return to the progenitor 3^{⋆10}:_{m} M_{10} that gave rise to the image G = ⟨x, y, t⟩ ≅ 3_{2}·U_{4}(3) and append an automorphism s that commutes with N = ⟨x, y⟩ and inverts all of the symmetric generators. We obtain:

\frac{3^{⋆10}:_{m} (M_{10} × 2)}{[(1, 2, 3, 6, 5, 4, 7, X)(8, 9)t_{1}]^{5} = 1} ≅ 3_{2}·U_{4}(3):2′_{3},

where the 2′_{3} denotes that this is the version of 3_{2}·U_{4}(3):2_{3} with trivial centre. Let H = ⟨xs, ys⟩ be the diagonal M_{10} in ⟨x, y, s⟩ ≅ M_{10} × 2. Then t has just 10 images under H, t^{−1} not being among them. Thus both of the involutions st and ts have 10 images under H and so both ⟨xs, ys, st⟩ and ⟨xs, ys, ts⟩ are images of 2^{⋆10}:M_{10}. Let ξ = xs = sx, γ = ys = sy, τ_{1} = st and τ_{2} = ts. Then ξτ_{1} = xs.st = xt has order 5, as does ξτ_{2} = sx.ts = (xt)^{s}. Therefore, both ⟨xs, ys, st⟩ and ⟨xs, ys, ts⟩ can be seen to satisfy the presentation for L_{3}(4):2_{1} shown above and thus ⟨xs, ys, st⟩ ≅ ⟨xs, ys, ts⟩ ≅ L_{3}(4):2_{1}. A permutation action of ⟨x, y, t, s⟩ ≅ 3_{2}·U_{4}(3):2′_{3} over either of these subgroups gives rise to a faithful permutation representation of 3_{2}·U_{4}(3):2′_{3} of degree 486, which is the minimal degree of a faithful permutation representation of 3_{2}·U_{4}(3):2′_{3}.

Finally, we remark that N ≅ M_{10} has just two orbits on tetrads: 30 special tetrads, for example {1, 2, 3, 9}, and 180 non-special tetrads. Furthermore, the stabiliser of a special tetrad is S_{4} in its natural action on the points of the special tetrad. For m = 2 or 3, and the images 2·L_{3}(4).2^{2} (and also L_{3}(4):2_{1}) or L_{4}(3):2_{2} of m^{⋆10}:M_{10}, the subgroup ⟨t_{1}, t_{2}, t_{3}, t_{9}, Stab_{N}({1, 2, 3, 9})⟩ is visibly a homomorphic image of:

\frac{m^{⋆4}:S_{4}}{(2, 3) = t_{0}t_{1}t_{0}t_{1}} ,

which is PGL_{2}(7) or L_{3}(3) in the cases m = 2 or 3 respectively. In the case of the image 3_{2}·U_{4}(3) of 3^{⋆10}:_{m} M_{10}, we find that (1, 2, 3, 6, 5, 4, 7, X)(8, 9)t_{8} has order 4, that is t_{8}t_{9}^{−1}t_{8}t_{9}^{−1} = (1, 5)(2, 4)(3, 7)(6, X), and so ⟨t_{1}, t_{2}, t_{3}, t_{9}, Stab_{N}({1, 2, 3, 9})⟩ is a homomorphic image of:

\frac{3^{⋆4}:S_{4}}{(2, 3) = t_{0}t_{1}^{−1}t_{0}t_{1}^{−1}} ≅ U_{3}(3) × 3,

and in fact we have that ⟨t_{1}, t_{2}, t_{3}, t_{9}, Stab_{N}({1, 2, 3, 9})⟩ ≅ U_{3}(3). (The progenitor 3^{⋆4}:S_{4} is the one with permutation action—even in the latter case.)

8.3.2 A presentation of 3^{4}:M_{10}

We now turn our attention to the true-monomial progenitor of form 3^{⋆10}:M_{10}. This progenitor is essentially unique. (There are two linear representations of M_{9} that are not trivial on M_{9} ∩M_{10}′ and these are interchanged by an outer automorphism of M_{10}.) Then:

G = \frac{3^{⋆10}:_{m} M_{10}}{[(1, 3, 5, 7)(2, 6, 4, X)t_{1}]^{4} = 1} ≅ 3^{4}:M_{10}.

The extra relation here is (πt_{i})^{4} = 1 where π ∈ M_{10}′ has cycle-type 4^{2}.1^{2} and i is in one of the 4-cycles of π. (It does not matter which one, for π commutes with an element of cycle-type 8.2.) Naturally, all such πt_{i} have order 4. This group appears in McL as ⟨t_{10}, N_{0}⟩. This group does not appear in 3·McL, for it lifts to a 3^{1+4}:M_{10} in 3·McL.

8.3.3 A presentation of 5{1+2 \atop +}:3

Using Magma, we found the following presentation for 5{1+2 \atop +}:3:

G = ⟨v, y | v^{3} = y^{5} = 1 = vyvy^{2}v^{−1}yv^{−1}y^{2}⟩.

It is of interest to note that we may apply an automorphism of order 2 centralising y and inverting v to obtain 5{1+2 \atop +}:S_{3}. In the progenitor 3^{⋆110}:_{m} M_{11}, when v = t_{23} and y = (1, 3, 9, 5, 4)(2, 6, 7, X, 8), the relation vyvy^{2}v^{−1}yv^{−1}y^{2} is the main relation required to define the McLaughlin group.

8.4 The McLaughlin group itself

8.4.1 The ‘basic’ presentation

We found that if we add sufficient relations to 3^{⋆110}:_{m} M_{11} to define:

(i) ⟨t_{01}, N_{0}⟩ ≅ 3_{2}·U_{4}(3),
(ii) ⟨t_{10}, N_{0}⟩ ≅ 3^{4}:M_{10},
(iii) ⟨t_{23}, (1, 3, 9, 5, 4)(2, 6, 7, X, 8)⟩ ≅ 5{1+2 \atop +}:3,
and (iv) [t_{01}, t_{10}] = 1,

we obtain a presentation for McL. Presentations for each of these subgroups were given in Section 8.3. The resulting presentation of McL is realised below:

\begin{matrix}⟨x, y, t | x^{8} = & y^{5} = y^{x^{2}}y^{−2} = 1 = (xy)^{6} = (x^{3}y)^{11} = t^{3} = t^{x^{2}}t = [t, t^{x}] \\ = & [t, xyxyxy^{2}xy] = (t^{x}x^{yxy^{−1}})^{5} = (t^{xy}xyxyxy^{2}xy)^{4} = vyvy^{2}v^{−1}yv^{−1}y^{2}⟩, \end{matrix}

where x ∼ (0, 1)(2, 3, 6, 9, X, 4, 7, 5), y ∼ (1, 3, 9, 5, 4)(2, 6, 7, X, 8) and t ∼ t_{01} and v ∼ t_{23} is shorthand for the element t^{y^{−1}xy^{−2}x}.

8.4.2 When do we have ⟨t_{ij}, π⟩ ≅ 5{1+2 \atop +}:3?

At this moment, we pause to describe how one may arrive at the subgroups described in (iii) above. In Section 8.6.1, we describe, given a symmetric generator, t_{ij}, how to obtain π ∈ N such that ⟨t_{ij}, π⟩ ≅ 5{1+2 \atop +}:3. The subgroup ⟨π⟩, which is cyclic of order 5, is uniquely determined. Conversely, given π ∈ N such that o(π) = 5, the t_{ij}^{±1} such that ⟨t_{ij}, π⟩ ≅ 5{1+2 \atop +}:3 is unique up to conjugacy by a power of π. We describe how to obtain such a t_{ij} given π below.

Let π = (p_{1}, p_{2}, p_{3}, p_{4}, p_{5})(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) where {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}} is a special pentad of S and {n_{1}, n_{2}, n_{3}, n_{4}, n_{5}} is not. Then we may suppose that we have chosen our notation so that ⟨π⟩ is normalised by σ = (p_{2}, p_{3}, p_{5}, p_{4})(n_{2}, n_{3}, n_{5}, n_{4}) ∈ N. Then our required symmetric generator is t_{p_{1}n_{1}}. As an example, we consider π = y = (2, 6, 7, X, 8)(1, 3, 9, 5, 4), where {2, 6, 7, X, 8} is a pentad of S and {1, 3, 9, 5, 4} is not. Then π is normalised by (x^{y})^{2} = (6, 7, 8, X)(1, 4, 9, 5). Thus we find that t_{23} is the symmetric generator we seek.

8.4.3 Variants of the basic presentation

In the basic presentation of Section 8.4.1 (see above), it turns out that [t, t^{x}] = 1 and (t^{x}x^{yxy^{−1}})^{5} = 1 are redundant. This may be verified by coset enumeration over ⟨t_{10}, N_{0}⟩ ≅ 3^{4}:M_{10}. Upon replacing (t^{xy}xyxyxy^{2}xy)^{4} = 1 by (t^{x}x^{yxy^{−1}})^{5} = 1 we obtain a presentation for 3·McL. The order may be verified by coset enumeration over ⟨t_{01}, N_{0}⟩ which is a homomorphic image of 3_{2}·U_{4}(3), though the presence of the central 3 is less clear.

Writing v for t_{23}, we obtain presentations:

\begin{matrix}⟨x, y, v | x^{8} = & y^{5} = y^{x^{2}}y^{−2} = 1 = (xy)^{6} = (x^{3}y)^{11} = v^{3} = [v, x^{2}y^{−1}] \\ = & v^{x^{−1}y^{2}xyxy^{−2}x}v = (vx^{−1}y^{−2}xy^{2})^{4} = vyvy^{2}v^{−1}yv^{−1}y^{2}⟩ ≅ McL, \end{matrix}

and

\begin{matrix}⟨x, y, v | x^{8} = & y^{5} = y^{x^{2}}y^{−2} = 1 = (xy)^{6} = (x^{3}y)^{11} = v^{3} = [v, x^{2}y^{−1}] \\ = & v^{x^{−1}y^{2}xyxy^{−2}x}v = (vy^{−2}xy^{−1}x)^{5} = vyvy^{2}v^{−1}yv^{−1}y^{2}⟩ ≅ 3·McL. \end{matrix}

In fact, in the last case, the relation (vy^{−2}xy^{−1}x)^{5} = 1 is redundant. However, on its omission, the biggest subgroup that we know that we have a presentation for is the control subgroup N ≅ M_{11}. Therefore we have to verify the order of G by coset enumeration over N. It took a coset table with some 8 000 000 rows to verify that N did indeed have the required index of 340 200.

8.5 A presentation for McL:2 (and 3·McL:2)

Since McL has an outer automorphism of order 2, it seems natural to ask if we may also obtain a presentation for McL:2. In fact, this turns out to be particularly easy, for the centraliser in McL of s, an outer automorphism of order 2, is M_{11}. So we take s to centralise x and y. Now s cannot centralise t, but perhaps conjugation by s can invert t (and hence invert all of the symmetric generators). Indeed, we were careful to show that all the relations we added also held after applying the map (x, y, t) ↦ (x, y, t^{−1}), and so the proposed conjugation by s is an automorphism. Since s inverts t, s is non-trivial, and since no non-trivial element of McL can centralise M_{11}, it follows that this automorphism must be an outer automorphism. The map (x, y, t) ↦ (x, y, t^{−1}) is also an automorphism when our image is 3·McL and adding s (defined as above) in this case yields the group 3·McL:2.

8.6 Subgroups of McL generated by subsets of the symmetric generators

Below, we tabulate all subgroups of McL generated by 2 symmetric generators (up to conjugacy). We then give all extensions of these subgroups by extra symmetric generators which do not generate the whole of G. Recall that M_{11} is 4-transitive on 11 points.

8.6.1 The case ⟨t_{01}, t_{23}⟩

We have that ⟨t_{01}, t_{23}⟩ ≅ 5{1+2 \atop +}:3 contains t_{48}, t_{67} and t_{X9} as well as t_{01} and t_{23}. Any other symmetric generator along with ⟨t_{01}, t_{23}⟩ generates the whole of G. Let S be the Steiner system S(4, 5, 11) preserved by N. Then we find the symmetric generators in ⟨t_{ij}, t_{kl}⟩ as illustrated below with the case ⟨t_{01}, t_{23}⟩. Firstly, find the unique permutations whose partial permutations on {0, 1, 2, 3} are (0)(1)(2, 3), (0, 1)(2)(3), (0, 1)(2, 3) and (0)(2)(1, 3).

These are shown below and can easily be found using Curtis’s ‘Kitten’ [12].

\begin{matrix}α = & (0)(1)(2, 3)(4, 6)(5, 8)(7, X)(9) \\ β = & (0, 1)(2)(3)(4, 6)(5, 7)(8, X)(9) \\ γ = αβ = & (0, 1)(2, 3)(4)(5, X)(6)(7, 8)(9) \\ δ = & (0)(2)(1, 3)(4, 8)(5, X)(6, 7)(9) \end{matrix}

Now 9 completes S_{0123} = {0, 1, 2, 3} to a special pentad of S. Let i and k be the other two fixed points of γ (so WLOG i = 4 and k = 6). Let (i, j) and (k, l) be the cycles of δ containing i and k respectively. Then t_{ij} and t_{kl} are two of the extra symmetric generators we are seeking—in this case they are t_{48} and t_{67}. The last symmetric generator is t_{X9} or t_{59} where (5, X) is the other cycle of δ outside S_{0123} and 9 is the fixed point of δ outside S_{0123}. To decide which one it is, take the four i’s of the known t_{ij} that occur in our subgroup, i.e., take the set {0, 2, 4, 6}, and extend to a pentad {0, 2, 4, 6, X} of S. This last symmetric generator is then t_{X9}.

We must now find an element, π, of order 5 in N that cycles these 5 symmetric generators. Without loss of generality, π has the form (5)(0, 2, …)(1, 3, …). Moreover, it is inverted by the unique involution, (2, 6)(3, 7)(4, X)(8, 9), that fixes 0, 1 and 5. Therefore π has the form (5)(0, 2, …, 6)(1, 3, …, 7) and since π cycles these 5 symmetric generators there are only two possibilities for π and they are (5)(0, 2, 4, X, 6)(1, 3, 8, 9, 7) and (5)(0, 2, X, 4, 6)(1, 3, 9, 8, 7). It turns out that π = (5)(0, 2, 4, X, 6)(1, 3, 8, 9, 7). (The other “possibility” is not an element of N.)

This long-winded and rather unattractive method of finding the extra symmetric generators has a much more natural description when we come to consider Ly as a homomorphic image of 3^{⋆2520}:_{m} 2·A_{11} in Chapter 10.

8.6.2 The case ⟨t_{01}, t_{12}⟩

We have that ⟨t_{01}, t_{12}⟩ ≅ SL_{2}(5) contains t_{01} and t_{12} and no other symmetric generator. The following table lists the subgroups generated by subsets of {t_{01}, t_{10}, t_{02}, t_{20}, t_{12}, t_{21}} up to N-conjugacy that contain a conjugate of ⟨t_{01}, t_{12}⟩, but which do not generate the whole of ⟨t_{01}, t_{10}, t_{02}, t_{20}, t_{12}, t_{21}⟩ ≅ 2·A_{8}. The first column lists the subgroup generators, the second lists the subgroup generated by these generators and the third column lists all the symmetric generators in this subgroup.

Generators Subgroup t_{ij} present
t_{01}, t_{12} SL_{2}(5) t_{01}, t_{12}
t_{01}, t_{12}, t_{21} SL_{2}(5) × 3 t_{01}, t_{12}, t_{21}
t_{01}, t_{12}, t_{10}  2·A_{6}  t_{01}, t_{10}, t_{02}, t_{12}
t_{01}, t_{12}, t_{02}  2·A_{6}  t_{01}, t_{10}, t_{02}, t_{12}
t_{01}, t_{10}, t_{02}, t_{12}  2·A_{6}  t_{01}, t_{10}, t_{02}, t_{12}
t_{01}, t_{12}, t_{20}  2·A_{7}  t_{01}, t_{12}, t_{20}

If t_{ij} is a symmetric generator not in {t_{01}, t_{10}, t_{02}, t_{20}, t_{12}, t_{21}} and j ≠ 1 then we have ⟨t_{01}, t_{12}, t_{ij}⟩ = G. For i ⩾ 3 we have that ⟨t_{01}, t_{12}, t_{i1}⟩ ≅ 3{1+4 \atop +}:2·A_{5} and contains just the 11 symmetric generators t_{j1} (for all j ≠ 1) and t_{12}.

8.6.3 The case ⟨t_{01}, t_{02}⟩

We have that ⟨t_{01}, t_{02}⟩ ≅ SL_{2}(3) contains t_{01} and t_{02} and no other symmetric generator. The only groups generated by symmetric generators that contain ⟨t_{01}, t_{02}⟩ but which have no subgroups of the form ⟨t_{ij}, t_{jk}⟩ or ⟨t_{ij}, t_{kl}⟩ are subgroups of ⟨t_{01}, t_{02}, t_{03}, …, t_{0X}⟩ ≅ U_{4}(3), which contains no symmetric generator except the t_{0i}. We have that ⟨t_{0i}, t_{0j}, t_{0k}, t_{0l}⟩ ≅ U_{4}(3) except when {0, i, j, k, l} is a special pentad of the S(4, 5, 11) S in which case we have ⟨t_{0i}, t_{0j}, t_{0k}, t_{0l}⟩ = ⟨t_{0i}, t_{0j}, t_{0k}⟩ ≅ U_{3}(3). For example, we have ⟨t_{01}, t_{02}, t_{03}⟩ = ⟨t_{01}, t_{02}, t_{03}, t_{09}⟩ ≅ U_{3}(3) whereas ⟨t_{01}, t_{02}, t_{03}, t_{0i}⟩ ≅ U_{4}(3) for i ∈ {4, 5, 6, 7, 8, X}.

8.6.4 The case ⟨t_{10}, t_{20}⟩

We have that ⟨t_{10}, t_{20}⟩ ≅ 3^{2} contains t_{10} and t_{20} and no other symmetric generator. The subgroups generated by symmetric generators not already dealt with are all subgroups of ⟨t_{10}, t_{20}, …, t_{X0}⟩ ≅ 3^{4}. We have that ⟨t_{10}, t_{20}, t_{30}⟩ = ⟨t_{10}, t_{20}, t_{30}, t_{90}⟩ ≅ 3^{3} and that ⟨t_{10}, t_{20}, t_{30}, t_{i0}⟩ ≅ 3^{4} for i ∈ {4, 5, 6, 7, 8, X}.

8.6.5 The case ⟨t_{01}, t_{10}⟩

We have that ⟨t_{01}, t_{10}⟩ ≅ 3^{2} contains t_{01} and t_{10} and no other symmetric generator. All the subgroups ⟨t_{01}, t_{10}, t_{ij}⟩ and so on have already been dealt with in one of the subsections above.

8.7 Some maximal subgroups of McL and McL:2

In this section we give a representative of each conjugacy class of maximal subgroups of McL and McL:2. We give generators for such a subgroup both in terms of x, y and t and in terms of the symmetric generators. In some cases, we also give a brief description of how we obtained the group. It is usually the case that if a maximal subgroup of McL extends to a maximal subgroup of McL:2 then our given representative(s) is/are normalised by s (an outer automorphism which has order 2, centralises x and y and inverts t).

The outer automorphism, s, of McL that commutes with x and y while inverting t either normalises each group in the above list, or if not, swaps it with another member in the above list, with the exception that s does not normalise ⟨t_{03}^{−1}t_{12}, t_{23}, t_{32}⟩ ≅ U_{4}(3).

U_{4}(3) and U_{4}(3):2_{3}

This is ⟨t_{0i}⟩ = ⟨t_{01}, N_{0}⟩ = ⟨t, x^{2}, y, xyxyxy^{2}xy⟩ = ⟨t, y⟩ and was used to help obtain a presentation for McL. We also found that ⟨t_{03}^{−1}t_{12}, t_{23}, t_{32}⟩ ≅ U_{4}(3), where one of the generators t_{23} or t_{32} is redundant and t_{03}^{−1}t_{12} has order 5 (not 15). The first of these U_{4}(3)’s is normalised by s and we obtain ⟨t, x^{2}, y, xyxyxy^{2}xy, s⟩ = ⟨t, y, s⟩ ≅ U_{4}(3):2_{3}. The second of these subgroups is not normalised by s.

M_{22} (both classes)

We found that ⟨t_{03}^{−1}t_{12}, t_{10}t_{01}⟩ = ⟨t^{−y}t^{y^{−2}x}, t^{x}t⟩ ≅ M_{22}. Apply the outer automorphism (s) of G that fixes x and y while inverting t to get ⟨t_{03}t_{12}^{−1}, t_{10}t_{01}⟩ = ⟨t^{y}t^{−y^{−2}x}, t^{x}t⟩ ≅ M_{22} but not conjugate to the M_{22} already obtained. Of course, these two M_{22}’s are conjugate in Aut(G) ≅ McL:2.

U_{3}(5) and U_{3}(5):2

We found that ⟨y, t^{x}t⟩ = ⟨(1, 3, 9, 5, 4)(2, 6, 7, X, 8), t_{10}t_{01}⟩ ≅ U_{3}(5). We found that this group also contained x^{−1}y[x, y]x ∼ (0, 3, 1, 4)(2, 7, 6, X). This group is normalised by s and we obtain ⟨y, t^{x}t, s⟩ ≅ U_{3}(5):2.

3{1+4 \atop +}:2·S_{5}^{−} and 3{1+4 \atop +}:4·S_{5}

This is N(3A) in McL. Taking 3A to be t_{10}, we adjoined everything we could think of that normalised ⟨t_{10}⟩ and this was sufficient to generate the whole of 3{1+4 \atop +}:2·S_{5}^{−}. Namely, we have ⟨t_{01}, t_{i0}, N_{01}⟩ = ⟨t, t^{x}, t^{xy}, x^{2}, xyxyxy^{2}xy⟩ ≅ 3{1+4 \atop +}:2·S_{5}^{−}. This subgroup is normalised by s.

3^{4}:M_{10} and 3^{4}:(M_{10} × 2)

This is ⟨t_{10}, N_{0}⟩ = ⟨t^{x}, x^{2}, y, xyxyxy^{2}xy⟩ and was used to help obtain a presentation for McL. This subgroup is normalised by s.

L_{3}(4):2_{2} and L_{3}(4):2^{2}

We found that ⟨y, [x, t]^{y^{−1}xy^{−1}}⟩ = ⟨(1, 3, 9, 5, 4)(2, 6, 7, X, 8), t_{62}^{−1}t_{26}⟩ ≅ L_{3}(4). Adjoining x^{−1}y[x, y]x ∼ (0, 3, 1, 4)(2, 7, 6, X) to this subgroup yielded L_{3}(4):2_{2}. This subgroup is normalised by s.

2·A_{8} and 2·S_{8}^{−}

The particularly nice version of this group is ⟨t_{01}, t_{02}, t_{10}, t_{12}, t_{20}, t_{21}⟩. We also found that ⟨t_{03}t_{30}, t_{21}^{−1}t_{12}⟩ = ⟨t^{y}t^{xy}, [x, t]^{y^{−2}x}⟩ ≅ 2·A_{8}. Both of these subgroups are normalised by s.

2^{4}:A_{7} (both classes)

We found that ⟨t_{03}^{−1}t_{12}, t_{46}t_{64}⟩ = ⟨t^{−y}t^{y^{−2}x}, (tt^{x})^{y^{−1}xy^{−1}}⟩ ≅ 2^{4}:A_{7}. We obtain a representative of the other McL conjugacy class of 2^{4}:A_{7} by applying s to the 2^{4}:A_{7} we have already obtained. Thus, we have that ⟨t_{03}t_{12}^{−1}, t_{46}t_{64}⟩ = ⟨t^{y}t^{−y^{−2}x}, (tt^{x})^{y^{−1}xy^{−1}}⟩ ≅ 2^{4}:A_{7}. Of course, these two subgroups are conjugate in McL:2.

M_{11} and M_{11} × 2

The canonical example of a subgroup from this class is the control subgroup N = ⟨x, y⟩. This group is normalised, indeed centralised, by s and we get ⟨x, y, s⟩ ≅ M_{11} × 2.

5{1+2 \atop +}:3:8 and 5{1+2 \atop +}:3:(8:_{5}2)

It was found that ⟨t_{23}, y, x^{2}, x^{t_{80}}⟩ ≅ 5{1+2 \atop +}:3:8. This subgroup contained t_{80}xt_{80}^{−1} as well as t_{80}^{−1}xt_{80}. We have that y ∼ (1, 3, 9, 5, 4)(2, 6, 7, X, 8), x^{2} ∼ (2, 6, X, 7)(3, 9, 4, 5) and x ∼ (0, 1)(2, 3, 6, 9, X, 4, 7, 5) along with t_{23} = t^{y^{−1}xy^{−2}x} and t_{80} = t^{yxy^{2}xyx}. This group is normalised by s.

McL

This is a triviality in the sense of the Atlas [10, page xix]. Of course, we have that ⟨x, y, t⟩ ≅ McL.

2^{2|4}:(S_{3} × S_{3})

This is a novelty in the sense of Wilson [38] (and also the Atlas [10, page xix]). We did actually find this subgroup, though our generators for it can hardly be said to be pretty. We proceeded as follows. Let a = t_{01}t_{10} = tt^{x}, b = t_{37}t_{69} = t^{yxy}t^{y^{3}xyx}, c = t_{68}t_{X4} = t^{xy^{2}xyx^{5}y^{2}}t^{yxy^{2}x} and π = sx^{2}yxyxy^{2}xy = s × (0, 1)(3, X)(4, 7)(8, 9). Then the involution π commutes with a and b^{π} = c^{−1}. Let H_{1} = ⟨a, b⟩, H_{2} = ⟨a, c⟩ and H_{12} = H_{1} ∩ H_{2}. Then H_{1} ≅ H_{2} ≅ 2^{4}:A_{7}, H_{2} = H_{1}^{π} and H_{12} ≅ 2^{4}:(A_{4} × 3):2 (of order 1152 = 2^{7}.3^{2}) is normalised by π. We found that H_{12} = ⟨a, bab^{−1}, b^{2}a^{−1}b^{−2}ab, b^{2}aba^{−1}b^{2}ab⟩. Thus,

M = ⟨H_{12}, π⟩ = ⟨a, bab^{−1}, b^{2}a^{−1}b^{−2}ab, b^{2}aba^{−1}b^{2}ab, π⟩,

is the required maximal subgroup of shape 2^{2|4}:(S_{3} × S_{3}) in McL:2.

Please note that we use the notation p^{n|m} to denote a special group of order p^{n+m} with a centre (and derived subgroup) of order p^{n}. Also, McL:2 has more than one conjugacy class of subgroups of order 2^{8}.3^{2}. We have executed some checks (such as orbits sizes and centraliser orders) to make sure that we do have the correct (i.e., maximal) subgroup.

Constructing the Harada–Norton group HN

9.1 Introduction

In 1973, Bernd Fischer found evidence for the Monster simple group (and also the Baby Monster). The Monster was discovered independently by Robert L. Griess at about the same time. For g ∈ M (the Monster), C_{M}(g) often contains a unique non-abelian composition factor, and in many cases this composition factor is a sporadic simple group. For example, when g ∈ 2A, 2B, 3A, 3B, 5B, 7A, 7B, 11A or 13A this composition factor is B, Co_{1}, Fi′_{24}, Suz, J_{2}, He, A_{7}, M_{12} or L_{3}(3) respectively. These groups were all known at the time. However, C_{M}(3C) ≅ 3×Th and C_{M}(5A) ≅ 5×HN had a non-abelian (sporadic) composition factor whose existence was previously unknown. Koichiro Harada deduced much information about HN from the knowledge of involution centralisers in this putative group, namely 2·HS:2 and 2{1+8 \atop +}.(A_{5} × A_{5}).2, and Simon Norton [30] constructed HN as a permutation group on 1 140 000 points and did much other work on the group.

Our aim is to construct HN as 133 × 133 matrices over ℚ(b5). When referring to this representation, the Atlas has a phrase, “Explicit matrices have been computed.” However, such an explicit construction does not seem to occur in the literature (such as Norton’s thesis), though it is possible that someone (such as Norton) has constructed explicit matrices previously. Inspection of Norton’s thesis [30] reveals that he [Norton] constructed 1 140 000 vectors which lay in a 133-dimensional subspace of a (1 + 462)- dimensional vector space over ℚ(b5) and specified the action of A_{12} (this was fairly easy) and an element g ∉ A_{12} (this was much harder) on the 1 140 000 vectors. In principle, we can convert the information in Norton’s thesis into 133 × 133 matrices generating HN but this would (probably) involve more work than we are proposing to spend on the construction. Moreover, this conversion is not of much mathematical interest and is prone to error. Norton constructed HN in the form HN = ⟨A_{12}, g⟩ whereas we propose to construct the group in the form HN = ⟨2·HS:2, t⟩—that is, as an amalgam of 2·HS:2 and (D_{10} × U_{3}(5))·2 over U_{3}(5):4 ≅ (U_{3}(5) × 2)·2. (Norton considered the amalgam of A_{12} and N_{HN}(A_{6}) ≅ (A_{6} × A_{6}).2^{2} over N_{A_{12}}(A_{6}) ≅ \frac{1}{2}(S_{6} × S_{6}).)

We also find a symmetric presentation for HN as a homomorphic image of a progenitor of shape 5^{⋆(176+176)}:_{m} 2·HS:2. This will be helpful, both in the construction of HN and for verifying that we have a representation of HN. Possible by-products are the permutation representations of HN of degrees 1 140 000 and 1 539 000 obtained by coset enumeration over A_{12} and 2·HS:2 respectively. (Though our implementation of Magma allows coset tables with sizes of several million, CosetAction appears to limit the coset table to size about 300 000. However, we can get around this limit by first using CosetSpace and applying CosetAction to the coset space.) Also, these permutations would take up a lot of memory. We give the approximate (and sometimes estimated) sizes of files required to store the various representations in Magma format.

The presentation: 749B.
Degree 132 matrices over 𝔽_{4}: 34.85kB per matrix.
Degree 133 matrices over 𝔽_{5}: 35.38kB per matrix.
Degree 133 matrices over ℚ(b5): 42kB per matrix.
Permutations on 1 140 000 points: 8.0MB per permutation (estimated).
Permutations on 1 539 000 points: 11.2MB per permutation (estimated).

In the case of the presentation, this is the size of the file and only some 300 bytes of the file is used to define the presentation. For the matrix files, we have only listed the (average) amount of space each matrix took up. In the case of matrices over 𝔽_{4} or 𝔽_{5}, this was invariant since each matrix entry occupied the same space. (We used W for w^2 in the 𝔽_{4} case.) In the ℚ(b5) case, this value is highly representation dependent, and in the representation above, several commonly occurring matrix entries were given shorter identifiers (which also made loading quicker). Of course, other formats might be more efficient, but equally they might not be.

9.2 The Higman–Sims group

We must first find a presentation for our proposed control subgroup 2·HS:2. (This is the group 2·HS.2 featured in the Atlas and not its isoclinic variant, which in any case is 2·HS·2.) Our starting point is the symmetric presentation of Curtis [16] given below:

\frac{2^{⋆50}:(U_{3}(5):2)}{t_{i}t_{k}t_{i}t_{j} = (i, k)} ≅ HS:2,

where i—j—k is a path in the Hoffman–Singleton graph, see Appendix C, and (i, k) is the unique element (involution) that fixes j, swaps i and k, and fixes the other five points joined to j.

So that we can successfully complete the required coset enumeration in HN, we have opted to create our own presentations of HS:2 based on the above symmetric presentation rather than use those given by Curtis [16]. We take:

⟨x, y | x^{2} = y^{4} = (xy)^{10} = (xyxy^{−1}xy^{2})^{2} = (xyxyxy^{−1}xy^{−1})^{4} = (xyxyxyxy^{2})^{7} = 1⟩, as a presentation for the control subgroup H ≅ U_{3}(5):2 in which we find (using the LowIndexSubgroups facility on Magma) that H_{0} = ⟨x, y^{2}, yxyxy^{−1}xy^{−1}x⟩ ≅ S_{7} (has index 50 in H). Adjoining z such that z^{2} = (zx)^{2} = [z, y^{2}] = [z, yxyxy^{−1}xy^{−1}x] = 1 gives us a presentation for the progenitor 2^{⋆50}:(U_{3}(5):2) where z is a symmetric generator, and adjoining yxyxy^{−1}xy^{−1} = z^{yxy^{−1}x}zz^{yxy^{−1}xyxyxy^{−1}xy^{−1}}z^{yxy^{−1}x}, which is equivalent to the relation ikij = (i, k) above, gives us HS:2. (Of course, we would like this extra relation to be shorter, but it will have to do and it has served its purpose.)

9.2.1 The double cover

We must now construct a presentation of the double cover. We shall now let x, y, z, H and N refer to entities in the double cover 2·HS:2 (Atlas version) and they will be preimages of what they were previously. Now the image of x in HS is a 2B involution so x must have order 4 and H = ⟨x, y⟩ ≅ (U_{3}(5) × 2)·2 which we shall often write as U_{3}(5):4. So x^{2} must be the central involution of 2·HS:2 = N. Without loss of generality, we may assume that y ∈ H′. Inspection of the classes of the elements y, xy, xyxy^{−1}xy^{2}, etc., (or merely looking at the number of x’s they contain and noting that H/H′ = ⟨xH′⟩) forces y^{4} = (xyxyxy^{−1}xy^{−1})^{4} = (xyxyxyxy^{2})^{7} = 1(= [x^{2}, y] = [x^{2}, x]) and (xy)^{10} = (xyxy^{−1}xy^{2})^{2} = x^{2}. (Insisting that y ∉ H′ merely changes (xyxyxyxy^{2})^{7} = 1 to (xyxyxyxy^{2})^{7} = x^{2}.) It turns out that

⟨x, y | y^{4} = (xy)^{10}x^{−2} = (xyxy^{−1}xy^{2})^{2}x^{−2} = (xyxyxy^{−1}xy^{−1})^{4} = (xyxyxyxy^{2})^{7} = 1⟩,

is a presentation for U_{3}(5):4 and forces x^{4} = [x^{2}, x] = [x^{2}, y] = 1. Now z is the preimage of an element of class 2C and so z must have order 2. Also z only commutes with an A_{7} × 2 of H, the commutator of z with an element in the outer half of (A_{7} × 2)·2 (the preimage of S_{7}) being the central involution x^{2}. Thus we have (zx)^{2} = 1, which accounts for both [z, x] = x^{2} and [z, x^{2}] = 1, and z^{2} = [z, y] = [z, yxyxy^{−1}xy^{−1}x] = 1. Either of the preimages of the last relation we used to define HS:2 can be used to define our group, so we pick yxyxy^{−1}xy^{−1} = z^{yxy^{−1}x}zz^{yxy^{−1}xyxyxy^{−1}xy^{−1}}z^{yxy^{−1}x}. This is because the map (x, y, z) ↦ (x^{−1}, y, z) fixes all the previous relations but interchanges the two possibilities here. This fact is not without significance. We may check that we have generated 2·HS:2 by taking a permutation representation of degree 1 408 over ⟨y, y^{x}⟩ ≅ U_{3}(5). Note that we shall use the notation N_{0} = H and N^{0} = H′.

9.3 The progenitor and extra relation

Since the Harada–Norton group contains subgroups isomorphic to 2·HS:2 (which we have called N) and (D_{10} ×U_{3}(5))·2 (which we have called T_{0}:N_{0}), it will come as no surprise to the reader that we shall consider a progenitor of the form 5^{⋆(176+176)}:_{m} 2·HS:2 in which all the non-trivial powers of t_{0} are conjugate under the action of N_{0} ≅ (U_{3}(5)×2)·2 ≅ U_{3}(5):4. Recalling the notation that for C ⩽ B we have Aut_{B}(C) = N_{Aut(B)}(C)/C_{Aut(B)}(C), we find that Aut_{N}(N_{0}) = Aut_{T_{0}:N_{0}}(N_{0}) = Inn(N_{0}) ≅ U_{3}(5):2 and that Aut(N_{0}) ≅ U_{3}(5):2 × 2.

Thus there are two (Aut_{N}(N_{0}), Aut_{T_{0}:N_{0}}(N_{0}))-double cosets of Aut(N_{0}) and hence two possible progenitors 5^{⋆(176+176)}:_{m} 2·HS:2 in which N_{0} acts faithfully on T_{0}. We do not know (at this point of the chapter) which one yields the Harada–Norton group, nor do we know a sensible way to distinguish them.

Thus, to obtain our progenitor, to our presentation of 2·HS:2 above we add t such that t^{5} = 1 = [t, y], t^{x} = t^{±2}. The two possibilities for the last relation give rise to non-isomorphic progenitors and we must determine in due course which one holds in the progenitor giving rise to the Harada–Norton group.

We must now consider what our extra relation can be. Now ⟨t, x^{2}, z⟩ centralises an A_{7} in H = ⟨x, y⟩ ≅ U_{3}(5). But ⟨t, x^{2}, z⟩ is generated by ⟨t, x^{2}⟩ ≅ D_{10} and ⟨x^{2}, z⟩ ≅ 2^{2} intersecting in ⟨x^{2}⟩ ≅ 2 and so we can only have ⟨t, x^{2}, z⟩ ≅ A_{5}. Replacing t by t^{2} if necessary, we may assume that zt has order 3.

For both of the progenitors 5^{⋆(176+176)}:_{m} 2·HS:2 in which we are interested, we factored out by the extra relation (zt)^{3} = 1 and enumerated the cosets of N ≅ 2·HS:2 in the resulting group G using Magma. In the case when t^{x} = t^{2} the coset table collapsed before reaching a size of 30 000 cosets to obtain |G : N| = 1. (We used a coset table with a potential size of 200 000 (cosets) and Hard:=true here.) This was enough to show that G = 1 in this case. In the case t^{x} = t^{−2} and with the redundant relation x^{4} = 1 and using a coset table of size at most 2 000 000 with Hard:=true, we found that |G : N| = 1 539 000 after some 1 600 seconds CPU time (on fourier) and maximum of 1 773 876 rows of the coset table. (The coset enumeration with the same parameters and run on the same machine as in the previous sentence but without using the relation x^{4} = 1 took about 2 000 seconds CPU time and had a maximum of 1 787 845 rows of the coset table.)

9.3.1 The automorphism group

We shall now consider possible outer automorphisms of HN. Let a be such an automorphism. We know that HN has just one class of 2·HS:2 (for such groups must be the centralisers of 2A involutions) and so we may assume that a normalises N = ⟨x, y, z⟩. Furthermore, we know that Aut(2·HS:2) ≅ HS:2 × 2 and so, multiplying by a suitable inner automorphism, we may assume that [x, a] = [y, a] = 1 and that z^{a} = z or zx^{2}. But now a centralises N^{0} and so a normalises C_{HN}(N^{0}) = ⟨t⟩. So t^{a} ∈ {t, t^{2}, t^{−2}, t^{−1}}. Now zt, zt^{−1}, zx^{2}t^{2} and zx^{2}t^{−2} have order 3 and zt^{2}, zt^{−2}, zx^{2}t and zx^{2}t^{−1} have order 5. Thus if z^{a} = z then t^{a} = t or t^{−1} and (conjugation by) a is realised as conjugation by 1 or x^{2} respectively, so a is inner. Therefore z^{a} = zx^{2} and t^{a} = t^{2} or t^{−2}, so a has order 4 and squares to (conjugation by) x^{2}. Replacing a by a^{−1} if necessary, we may assume that t^{a} = t^{2}. Note that za has order 2 so that our extension of HN to HN.2 is split.

In summary, to obtain Aut(HN) ≅ HN:2 from HN, we adjoin a such that a^{2} = x^{2}, [a, x] = [a, y] = 1, z^{a} = zx^{2} and t^{a} = t^{2}.

9.3.2 Some subgroups of HN and HN:2

Unfortunately, we have not found many of the maximal subgroups of HN owing to a number of factors including the size of the representation in which we are required to work (the minimal degree of a matrix representation is 132 over 𝔽_{4}) and the fact that the only subgroups generated by subsets of the symmetric generators are A_{n} (5 ⩽ n ⩽ 12), 5{1+2 \atop +}, U_{3}(5), 2·HS and HN (and the trivial cases 1 and 5). However, we have found A_{12}, 2·HS:2, (U_{3}(5) × D_{10})·2 and (A_{6} × A_{6})·D_{8} and we give generators for these subgroups below.

A_{12} ⟨H_{0}, z_{0}, t_{0}, z_{1}⟩ ⟨x, y^{2}, yxyxy^{−1}xy^{−1}, z, t, z^{yxy}⟩
2·HS:2  N  ⟨x, y, z⟩
2{1+8 \atop +}.(A_{5} × A_{5}).2  see below  see below
(U_{3}(5) × D_{10})·2 T_{0}:N_{0} = ⟨N_{0}, t_{0}⟩  ⟨x, y, t⟩
(A_{6} × A_{6})·D_{8}  see below  see below

where z_{0} commutes with H′ = N′_{0} and z_{1} commutes with an A_{6} of H′_{0} (and in fact both z_{0} and z_{1} commute with the central involution of H_{0}). Each of these subgroups (including those whose generators we give below) is normalised by a and adjoining a to each of these subgroups yields S_{12}, 4·HS:2, 2{1+8 \atop +}.(A_{5}×A_{5}).2^{2}, 5:4×U_{3}(5):2 and (S_{6}×S_{6}):2^{2} respectively.

In fact, two (distinct cyclic subgroups generated by) symmetric generators from the same 176-orbit of 2·HS generate a copy of A_{6} and this subgroup contains two symmetric generators (the mates of the first two) in the second 176-orbit of 2·HS. We find that this extends quite naturally to A_{12} as follows. Take a symmetric generator, t say. Then the stabiliser of t has orbits 50 + 126 on the 176-orbit of 2·HS to which t does not belong. Moreover, the 50-orbit can be given the structure of the Hoffman–Singleton graph (see Curtis [16] to see how to do this) and we pick a symmetric generator u in this 50-orbit, along with its neighbours u_{1}, …, u_{7} in this 50-orbit regarded as being a Hoffman–Singleton graph. The group containing t and the u’s contains altogether 8 symmetric generators from each 176-orbit and generates A_{12}. This A_{12} is actually an image of 5^{⋆(8+8)}:_{m} (A_{8}×2^{2}).2 with respect to the symmetric presentation as given in Section 4.3, namely:

\frac{5^{⋆(8+8)}:_{m} (A_{8} × 2^{2}).2}{(σt_{0})^{3} = ((0, 1)iσt_{0}^{2})^{3} = 1} ≅ A_{12}.

It can be shown that the extra relation ((zt)^{3} = 1) required to define HN is sufficient to show that ⟨x, y^{2}, yxyxy^{−1}xy^{−1}, z, t, z^{yxy}⟩ has a symmetric presentation as shown above. Thus we could have used coset enumeration over this subgroup to establish the order of G ≅ HN. In fact, coset enumeration proceeds quite smoothly over this subgroup.

We found an (A_{6} × A_{6})·D_{8} roughly as follows. We took two symmetric generators t_{0} and t_{1} in the same 176-orbit under 2·HS and found all elements of N that normalised both ⟨t_{0}⟩ and ⟨t_{1}⟩ or swapped them. We found that ⟨t_{0}, t_{1}, N_{01}, (0, 1)⟩ ≅ (A_{6} × A_{6}).2^{2} and normalised both A_{6}’s. Call this group H say. We then found an A_{12}, K say, that contained H′ ≅ A_{6} × A_{6}. Then N_{K}(H′) ≅ (A_{6} × A_{6}):2^{2} has some wreath action on the A_{6}’s and ⟨H, N_{K}(H′)⟩ is the required (A_{6} × A_{6})·D_{8}. In fact, we can take H = ⟨x, x^{y}, y^{2}xy^{2}xy^{2}, z, t⟩ and K = ⟨x, y^{2}, yxyxy^{−1}xy^{−1}, z, t, z^{yxy}⟩. Finally, upon defining d = (y^{x})^{2}(y^{xy^{2}})^{2}, f = yxyxy^{−1}xy^{−1}, e_{1} = (y^{z^{yxy}t})^{2}(y^{d^{−1}t})^{2}, e_{2} = e_{1}^{z^{yxy}t^{2}(xf)^{2}} and e_{3} = e_{2}^{z^{yxy}t^{2}((xy^{2})^{3}f^{−1})^{2}}, we find that ⟨H, e_{1}e_{2}e_{3}⟩ ≅ (A_{6} × A_{6})·D_{8}.

Inspection of the permutation character (here 1a^{+} ⊕1a^{−} ⊕175a^{+} ⊕175a^{−}) reveals that an element of class 2A of HS:2 fixes 16+16 points in the (176+176)-point action of HS:2 (the central involution of 2·HS:2 acts trivially). However for the monomial representation in which we are interested (whose complex character is 176ab) we have that elements in class +2A have trace 32 and that elements in class −2A have trace −32. Thus an element g in N-class +2A centralises an image of 5^{⋆(16+16)} and this image must be a subgroup of C_{HN}(g) = C_{HN}(2A) ≅ 2·HS:2 and in fact turns out to be 2·HS. An element in N-class −2A (for example −g = x^{2}g), which is in HN-class 2B, inverts all the symmetric generators in the same 5^{⋆(16+16)}. Let α = z and β = yxy^{−1}xyxy and let M = ⟨α, β⟩ ⩽ N. Then M ≅ 2.2{1+6 \atop +}.S_{5} has a centre generated by c_{1} = β^{4} ∈ +2A and c_{2} = (αβ)^{10} ∈ −2A and M has orbits (16+16)+(160+160) on the symmetric generators with t being in the (16+16)- orbit. Unsurprisingly, therefore, we found that ⟨α, β, t⟩ = C_{HN}(c_{1}) ≅ 2·HS:2. Thus we see that 2·HS:2 is a homomorphic image of a progenitor of shape 5^{⋆(16+16)}:_{m} 2.2{1+6 \atop +}.S_{5} and that HS:2 is a homomorphic image of 5^{⋆(16+16)}:_{m} 2{1+6 \atop +}:S_{5} though we have yet to investigate these configurations. (Note that M quotiented out by either ⟨x^{2}⟩ or ⟨c_{1}⟩ gives rise to isomorphic groups.) We note that Jabbar [26, pp. 133–134] has produced a presentation for HS as a homomorphic image of 5^{⋆(16+16)}:_{m} 4.2^{4}:S_{5} and this easily extends to a presentation of HS:2.

Using a standard centraliser-of-an-involution trick, we found that

C_{HN}(c_{2}) = ⟨α, β, (c_{2}(t^{zy}xz)^{3})^{6}⟩ ≅ 2{1+8 \atop +}.(A_{5} × A_{5}).2.

The trick we used was the ‘old’ trick as described in Section D.1.1, rather than the newer one described in Section D.1.2

9.4 The matrix construction

We now turn our attention to the 133-dimensional matrix construction of HN. The restriction to 2·HS:2 is as 77a^{+} ⊕ 56a^{±} where the two possible 56-dimensional representations are faithful and differ by only the sign of r5(= \sqrt{5}) and so we may choose either one. The 77-dimensional representation is really a representation of HS:2 and is a constituent of the 100 point permutation representation of HS:2 over ⟨y, xyxy^{2}xy^{−1}x, z⟩ ≅ M_{22}:2. (This latter group becomes 2·M_{22}:2 in 2·HS:2.) We used the characteristic zero MeatAxe facilities on Magma to chop this permutation module to obtain our required representation (and also the 22-dimensional representation while we were at it). Getting the 56-dimensional representation of 2·HS:2 was not so easy. Putting the representation of HN together was fairly straightforward. We describe these steps in separate (sub-)sections below.

9.4.1 The 56-dimensional representation of 2·HS:2

This is the hardest part of the whole construction. We find that representatives of the three conjugacy classes of A_{7} in N^{0} are given below:

\begin{matrix}M_{1} = & ⟨y^{2}, (y^{2})^{x}, xyxyxy^{−1}xy^{−1}⟩ \\ M_{2} = & ⟨y, xyxy^{2}xy^{−1}x⟩ \\ M_{3} = & ⟨y^{x}, yxy^{2}xy^{−1}x^{2}⟩ \end{matrix}

We note that x normalises M_{1} and that conjugation by x interchanges M_{2} and M_{3} (and the generators thereof). Thus the 2·M_{22}:2 above is ⟨M_{2}, z⟩.

The restriction of this representation to N_{0} is irreducible, and its restriction to N^{0} is 28b ⊕ 28c. Note that the central involution x^{2} acts as −I_{56}. The restrictions to M_{1}, M_{2} and M_{3} are as (14a ⊕ 14b) ⊕ (14a ⊕ 14b), (1a ⊕ 6a ⊕ 21a) ⊕ (14a ⊕ 14b) and (14a ⊕ 14b) ⊕ (1a ⊕ 6a ⊕ 21a) respectively.

Seed vectors for the two N^{0}-submodules may be obtained as the fixed points of M_{2} and M_{3} respectively. In fact, we may suppose that these fixed points v_{2} and v_{3} are related by v_{3} = v_{2}.x. Let M_{1} = ⟨m_{1}, m_{2}, m_{3}⟩ (generators as above). Then the seed vectors of 14a and 14b are the fixed vectors of ⟨m_{1}, m_{3}m_{1}m_{3}m_{1}m_{3}^{2}⟩ ≅ S_{5} and ⟨m_{1}, m_{3}m_{1}m_{3}m_{1}m_{3}^{2}m_{1}m_{3}⟩ ≅ L_{3}(2) respectively. We note that M_{1} is generated by m_{1} and m_{3}.

Firstly, we made our 56-dimensional representation of N_{0} ≅ (U_{3}(5) × 2)·2. This was done by inducing up the non-trivial linear representation of ⟨M_{2}, x^{2}⟩ ≅ A_{7} × 2 to obtain a monomial representation of degree 100 and chopping out the 56-dimensional constituent using the characteristic zero MeatAxe on Magma. We then found the 28-dimensional constituents of N^{0} ≅ U_{3}(5) by spinning up the seed vectors v_{2} and v_{3}. In each of these 28-dimensional constituents, we found the seed vectors for the constituents 14a and 14b as suggested in the paragraph above. These were labelled v_{2a}, v_{2b}, v_{3a} and v_{3b} respectively.

Spinning up each of these vectors under M_{1} gave a standard basis for M_{1} and a nearly standard basis for N^{0} and N_{0} by spinning up under M_{1} each of the vectors v_{2a}, v_{2b}, v_{3a} and v_{3b} in that order. (It is not standard since the seed vectors were only defined up to scalar multiplication.) When written with respect to this standard basis, matrices that centralise M_{1}, in particular the matrix of z, have the form:

[ \begin{matrix}αI_{14} & & βI_{14} \\ & ϵI_{14} & & ζI_{14} \\ γI_{14} & & δI_{14} \\ & ηI_{14} & & θI_{14} \end{matrix} ] ,

where the empty parts of the matrix consist of 0 entries. Using the fact that the representation was irreducible on ⟨N^{0}, z⟩ = ⟨U_{3}(5), z⟩ = N ≅ 2·HS:2 along with the fact that both z and zx had order 2 and trace 0 forced δ = −α, θ = −ϵ, γ = β and ζ = 4η along with α^{2} +β^{2} = 1 and ϵ^{2} +4η^{2} = 1 (at least when I did it—this is not reproducible without my matrices since I did not define the basis well enough to determine the exact matrix of x). Then knowledge of the expected traces of [z, y], (zyxyxy^{−1}x)^{2} and (zy^{−1}xy)^{2} (which are 2, 0 and 0 respectively) gave just four possibilities for (α, β, ϵ, η). In each case, the values of α, β, etc., lay in ℚ(r5) but not in ℚ. In fact, there were only two possibilities for (α, β, ϵ, η) up to field automorphisms of ℚ(r5). Finally, we substituted both possibilities into our chosen preimage of (i, k) = ikij to decide which one was correct. (In fact, the other possibility also gave rise to a representation of 2·HS:2, but the other preimage of (i, k) = ikij was satisfied in this case.)

9.4.2 The representation of HN

The restriction to N^{0} is as (21a ⊕ 28b ⊕ 28c) ⊕ (28b ⊕ 28c). We choose the seed vectors for the 28b and 28c contained in the 56-dimensional representation of N to be a fixed vector of M_{2} (v say) and vx (which is fixed by M_{3}) respectively. In the 77-dimensional representation of N we choose a fixed vector of M_{1} to be the seed vector of 21a. Now M_{8} = ⟨(xyxy^{2}xy)^{2}, (yxy^{2}xyx)^{2}⟩ ≅ 2·S_{5}^{+} has a fixed space of dimension 2 contained in the (28b ⊕ 28c) of N^{0}. We chose a fixed vector of M_{8}. This spun up to the whole of (28b ⊕ 28c). We chose our fixed points of the M_{2} and M_{3} in this subspace to be the seed vectors of 28b and 28c respectively. (Again, we ensured that the seeds v and v′ are related by vx = v′.)

When written with respect to this standard basis, N^{0} and x have the forms shown below:

N^{0} ∼ [ \begin{matrix}21a \\ & 28b \\ & & 28c \\ & & & 28b \\ & & & & 28c \end{matrix} ] and x ∼ [ X ; I_{28} ; I_{28} ; I_{28} ; −I_{28} ] ,

and N has block diagonal form 77 ⊕ 56. In fact, we had make sure that the manners in which we spun up 28b and 28c were ‘consistent’ to ensure that x had the special form described here (we spun up 28b under ⟨y, y^{x}⟩ and 28c under ⟨y^{x}, y⟩). Alternatively, we could simply define the standard basis of 28c to be {w_{1}x, …, w_{28}x} where {w_{1}, …, w_{28}} is the standard basis of 28b. Now the t that commute with N^{0} have the form:

t ∼ [ \begin{matrix}κI_{21} \\ & αI_{28} & & βI_{28} \\ & & ϵI_{28} & & ζI_{28} \\ & γI_{28} & & δI_{28} \\ & & ηI_{28} & & θI_{28} \end{matrix} ] ,

when written with respect to this basis. The equations t ≠ 1 = t^{5}, t^{x} = t^{3} have just two solutions (for t) up to elements centralising N and these differ only by the sign of r5. In fact, we find that κ = 1, α = δ = b/2, ϵ = θ = c/2, βγ = (−b − 3)/4, ζ = bβ and η = bγ where {b, c} = {b5, b5^{∗}} and without loss of generality we may set β = 1. However, our matrices for N are in a reasonably ‘nice’ form and the map induced by applying the field automorphism r5 ↦ −r5 to matrix entries fixes elements of N′ ≅ 2·HS and multiplies outer elements of N (i.e., those in N \ N′) by the central involution x^{2} of N. This map preserves N set-wise (and in fact is an automorphism of N) and so the two possibilities give rise to isomorphic groups (which must be HN). However, for only one of these possibilities do we have o(zt) = 3—for the other one we have o(zt^{2}) = 3 and o(zt) = 5.

9.5 Final remarks

In this chapter, we have dealt with the Harada–Norton group and constructed its smallest characteristic zero representation. In fact, it is not difficult show that 133 is the degree of a true representation of a progenitor of shape 5^{⋆(176+176)}:_{m} 2·HS.2 in which N_{0} acts faithfully on T_{0}.^{1} (By a true representation, we mean a representation giving rise to a true image in the sense of Definition 3.1.3.) This is also true of representations not in characteristics 2 or 5.

This is quite easy to see. Let ρ be a proper representation of 5^{⋆(176+176)}:_{m} 2·HS.2. The central involution of 2·HS.2 inverts t and so cannot be represented by a scalar matrix and so the restriction to N ≅ 2·HS.2 breaks up as the direct sum of nonzero representations ρ_{1} and ρ_{2} where ρ_{1} is a representation of HS:2 and ρ_{2} is faithful for 2·HS.2. The only faithful representations of 2·HS.2 of degree at most 133 have degree 56 and restrict to U_{3}(5) as 28b ⊕ 28c. Now the restriction of ρ_{1} to U_{3}(5) must contain at least 28b or 28c as a constituent, or else anything that commutes with N^{0} will also commute the central involution of N which should invert t, a contradiction. The smallest (irreducible) representation of HS:2 satisfying this condition is 77a^{±}. Thus any proper representation of 5^{⋆(176+176)}:_{m} 2·HS.2 has degree at least 56 + 77 = 133. We have constructed such a representation, so that 133 is indeed the smallest degree. This proof works without modification (but using the character tables in the ‘modular Atlas’ instead of the character tables in the Atlas) to obtain the corresponding result in every characteristic except 2 and 5.

Doing the corresponding thing in characteristic 2 must be sheer folly as we do not even know how to directly make faithful 2-modular representations of the control subgroup. The situation is not much better in characteristic 5. We shall restrict our attentions to the progenitors 5^{⋆(176+176)}:_{m} 2·HS:2. It is fairly clear that a proper 5-modular representation of such a progenitor cannot have degree less than 57. For the progenitor of this type that gave rise to the Harada–Norton group this is the minimal degree and we have constructed all 6 possible images and these include the affine group 5^{56}:2·HS:2 and a perfect irreducible group preserving a non-singular bilinear form which we presume is Ω_{57}(5). We have little idea what the minimal degree for the other progenitor is. However, we know that 77 is a

1This statement applies all four of them. There are four such progenitors since there are two groups of shape 2·HS.2 and two progenitors for each possible control subgroup.

lower bound (though this seems unlikely to be realised) and that 241 is an upper bound (probably much too high), for the image 5^{240}:2·HS:2 has a representation of this degree. We think that the minimal degree in this case is 111.

At the moment, we have our representation of HN over ℤ[b5, \frac{1}{2}, \frac{1}{5}]. We cannot eliminate the \frac{1}{2}’s from the representation without destroying the block diagonal form of 2·HS:2. For if we insist on block diagonal form for 2·HS:2, then the involution x^{2} reduced modulo 2 would become the identity, whereas it is not the identity in HN. We do not know whether can eliminate \frac{1}{5}’s from the representation while still preserving the block diagonal form of 2·HS:2. We decided to investigate the possibility of doing this. So we took Wilson’s 133-dimensional 𝔽_{5}-representation in [40] and restricted it to 2·HS:2 and found that it broke up as the direct sum 56 ⊕ 1 ⊕ 21 ⊕ 55. This means that the modulo 5 reduction of 77a would have to be as 1 ⊕ 21 ⊕ 55. For the integral representation of 77a we were using, the 5-modular reduction was uniserial. Though we have seen other rational versions of 77a whose 5-modular reduction is not uniserial, we have not yet seen a rational version of 77a that decomposes in the required manner, and nor do we know how to make one that does. Thus, we cannot eliminate \frac{1}{5}’s from the representation of HN while still preserving the block diagonal form of 2·HS:2 if we use an already-constructed 77a as one of those blocks. Even if we do manage to make a rational 77a whose 5-modular reduction has the required structure, this may still not be enough to eliminate the \frac{1}{5}’s.

Note however, that ℤ[b5] is a Euclidean domain, so that essentially the spin algorithm described in Parker’s integral MeatAxe paper [32] is sufficient to produce a ℤ[b5]-integral representation of HN. However, the integers involved would probably be very large (and very much ‘larger’ than the entries that one could expect to find in the corresponding 266-dimensional ℤ-integral representation of HN).

For the moment our generating matrices are not particularly palatable. A typical entry of a typical matrix of the group, when written as α + β\sqrt{5}, for α, β ∈ ℚ, would have the numerators and denominators of α and β being anything up to (and beyond?) 1 000 in modulus. (The generators themselves are much nicer.) We are working on ways to improve this situation. Most of the representations we have constructed have been over the algebraic integers. The exceptions are the (faithful) representations of HN and 2·HS:2, which we constructed over ℚ(b5).

The Lyons group

10.1 Introduction

In this chapter, we present the Lyons group Ly as a homomorphic image of the progenitor 3^{⋆2520}:_{m} 2·A_{11}. This is mentioned when dealing the McLaughlin group in Chapter 8 and the construction of McL in that chapter will be seen to be relevant here. We also mentioned this in Chapter 6 when we showed how to construct a 111-dimensional representation of Ly over 𝔽_{5} just by knowing that Ly was a homomorphic image 3^{⋆2520}:_{m} 2·A_{11} and the restriction of this representation to N ≅ 2·A_{11}.

10.1.1 The graphs

Firstly, we produce the bipartite graph of S_{11} (or A_{11}) acting on M_{11}’s with two M_{11}’s joined just when they intersect in M_{10}. (Two M_{11}’s that are joined lie in different A_{11} orbits.)

[figure: 1; 11; 110; 990; 55; 1584; 330; 495; 144; 1320; 11; 1; 10; 1; 9; 1; 1; 3; 9; 1; 8; 2; 1; 11; 8; 5; 8; 3; 1; 2; 1; 2; 5; 6]

From the bipartite graph above, we easily reproduce the graph below (on one of the orbits of M_{11} under A_{11}) where two points are joined just if they have distance 2 in the bipartite graph.

[figure: 1; 110; 330; 1584; 495; 0+8; 1+9; 35+25+10; 8+0+16; 110; 1; 9; 3; 5+10; 24+48; 3+24; 2+16; 5+15; 16+48; 72; 5; 9+9; 2+2]

Denote this second graph by Γ. Then we say that a, b ∈ Γ are orthogonal or perpendicular just if a and b are adjacent in Γ or a = b. Of course, we intend to identify the points of Γ with the (inverse pairs of) symmetric generators and then a and b are orthogonal just if their corresponding symmetric generators commute. We write a^{⊥} for the set of those points in Γ which are orthogonal to a. Similarly, if S is a subset of the vertices of Γ, then we write S^{⊥} for the set of vertices in Γ which are perpendicular to every vertex of S.

Let a and b be orthogonal vertices. Then their pointwise stabiliser fixes a unique third vertex c and c is orthogonal to both a and b. Such a triple of vertices is called a special triple. The pointwise stabiliser of a special triple is M_{9} and the setwise stabiliser is M_{9}:S_{3}. The preimage of the latter group in 2·A_{11} is 2×M_{9}:S_{3} since the Sylow 2-subgroup of M_{9}:S_{3} lies in a copy of M_{11}. (In symmetric generator terms, thinking about McL as an image of 3^{⋆110}:_{m} M_{11}, let a = t_{⋆}. Then the 110 vertices orthogonal to a may be labelled t_{ij} where i ≠ j with i, j ∈ {0, 1, …, 9, X}. Then b = t_{ij} and c = t_{ji} for some i and j.)

Let a and b be non-orthogonal vertices. Then we say that a and b are at 5{1+2 \atop +}:3 apart if b is in the orbit of size 1584 under the stabiliser of a (and vice-versa). Similarly, we say that a and b are at SL_{2}(3) apart if b is in the orbit of size 330 under the stabiliser of a and that a and b are at SL_{2}(5) apart if b is in the orbit of size 495 under the stabiliser of a. We have used this terminology because the symmetric generators corresponding to the above point pairs generate the groups named.

10.1.2 Some maximal subgroups

In this section we indicate how to find most of the maximal subgroups of Ly in terms of the symmetric generators. We ignore the Frobenius groups 67:22 and 37:18 as we have no hope of generating these in a nice manner (and little hope of finding them anyway). We indicate our progress on finding a representative of each of the other classes of maximal subgroups below. Any explicit calculations that we needed, for example to show that many of the groups we had were copies of G_{2}(5), were performed in the 111-dimensional representation of Ly which was constructed by Parker in 1985 and which we showed how to construct in Chapter 6. This representation is available in Wilson’s library [40].

G_{2}(5)

We have found several cases of ⟨π, t⟩ ≅ G_{2}(5) where π ∈ N and t is a symmetric generator. However, we have not investigated these configurations thoroughly enough to come up with a particularly nice way of generating this group.

3·McL:2

The centraliser of a 3A-element of Ly is 3·McL. Now a symmetric generator is in Ly-class 3A and so the subgroup a generated by a symmetric generator, t say, and the 110 other inverse pairs of symmetric generators that commute with t must generate a subgroup of 3·McL. In fact, they generate the whole of 3·McL. Without loss of generality, we may assume that t commutes with N^{0} and is inverted by outer elements of N_{0}. Then if τ(≠ t^{±1}) is a symmetric generator that commutes with t we have ⟨t, τ, N^{0}⟩ ≅ 3·McL and ⟨t, τ, N_{0}⟩ ≅ 3·McL:2. In both cases, t is a redundant generator.

5^{3}·L_{3}(5)

This is not easy to obtain for the symmetric generators are in Ly-class 3A whereas the elements of order 3 in 5^{3}·L_{3}(5) are in Ly-class 3B which may be obtained as a product of two commuting symmetric generators. We have not yet tried to locate this subgroup.

2·A_{11}

The canonical example of a group from this class is the control subgroup N. However, from our work on McL, we know that certain subsets of the symmetric generators generate 2·A_{8} and we surmise that 2·A_{11} (and other 2·A_{n}) can generated in this way also. We have seen that this is indeed the case by taking ‘random’ ⟨π, t⟩, where π ∈ N and t is a symmetric generator.

5{1+4 \atop +}:4·S_{6}

Let symmetric generators t and τ be at 5{1+2 \atop +}:3 apart in Γ. Then we can see from the diagram of Γ that t and τ have exactly 5 neighbours in common, that is {t, τ}^{⊥} contains 5 points. Then the 5 symmetric generators in \{t, τ\}^{⊥} and so can only generate 5{1+2 \atop +}:3 (both in total and pairwise). Thus {t, τ}^{⊥⊥} contains five points including t and τ. This gives an easy and natural way to complete a pair of this type to its natural pentad—unlike the McLaughlin group case.

We have that ⟨\{t, τ\}, \{t, τ\}^{⊥}⟩ ≅ 5{1+4 \atop +}:3^{2} < 5{1+4 \atop +}:4·S_{6}. We have not yet located enough elements to extend the 5{1+4 \atop +}:3^{2} to the whole 5{1+4 \atop +}:4·S_{6}.

3^{5}:(M_{11} × 2)

To obtain this group, take a subgroup M ≅ M_{11} × 2 in N such that M is not conjugate in N to N_{0}. (It is not conjugate in Ly either.) Then M has orbits of sizes 11 + 55 + 990 + 144 + 1320 on (inverse pairs of) symmetric generators. These orbits can easily be read off from the bipartite graph above. Let t be a symmetric generator in the 11-orbit. Then all such symmetric generators commute (since they are distance 2 in the bipartite graph) and generate a group 3^{5} which is normalised by M. Thus ⟨M, t⟩ ≅ 3^{5}:(M_{11} × 2).

3^{2|4}:2A_{5}.D_{8}

This is a normaliser of a special group of order 3^{6} with centre 3^{2}. It is also the normaliser in Ly of a 3^{2} of type 3A_{2}B_{2}. This 3^{2} can be generated by two mutually commuting symmetric generators, t and τ say. In 3^{⋆110}:_{m} M_{11}-parlance, we may take τ = t_{10} which is normalised by a subgroup H_{01} ≅ M_{9} of H = N^{0} ≅ M_{11}. We find that ⟨t_{10}, H_{10}, t_{01}, t_{i0}⟩ ≅ 3^{2|4}:2·S_{5}^{−}, contains t as a central element and that quotienting out by ⟨t⟩ gives the group 3{1+4 \atop +}:2·S_{5}^{−} familiar from the chapter on the McLaughlin group. Adding on the central involution v of N (which inverts both t and τ) extends the above group to a subgroup 3^{2|4}:4S_{5} of 3·McL:2. Let t = t_{0} and τ = t_{1} in the parlance of 3^{⋆2520}:_{m} 2·A_{11} (so that N_{0} ≅ N_{1} ≅ M_{11} × 2). Then N_{01} = N_{0} ∩ N_{1} ≅ M_{9} × 2 is normalised in N by a subgroup isomorphic to M_{9}:S_{3} × 2. This must therefore contain an involution σ that swaps ⟨t_{10}⟩ and ⟨t⟩. Adjoining this element to the 3^{2|4}:4S_{5} gives the full maximal subgroup. In summary, we have that ⟨t, t_{10}, H_{10}, t_{01}, t_{i0}, v, σ⟩ ≅ 3^{2|4}:2A_{5}.D_{8}.

10.1.3 The presentation

We shall now see what happens if we append the single relation πt_{23}π^{2}t_{23}πt_{23}^{−1}π^{2}t_{23}^{−1} = 1, where π = (1, 3, 9, 5, 4)(2, 6, 7, X, 8) ∈ N^{0}, to the progenitor 3^{⋆2520}:_{m} 2·A_{11}. The elements t_{ij} were defined in Section 10.1.1 and are the same as the elements t_{ij} in a certain subprogenitor 3^{⋆110}:_{m} M_{11}. We shall also assume that this relation holds in a proper image of 3^{⋆2520}:_{m} 2·A_{11}. (We can make the representation of 3^{⋆2520}:_{m} 2·A_{11} as 111 × 111 matrices over 𝔽_{5}, so this is routine, if somewhat tedious, to verify.)

Firstly, from Section 8.4 (in particular Section 8.4.3), we see that M_{1} := ⟨t_{01}, N_{0}⟩ ≅ 3·McL:2 (or McL:2). This means that ⟨t_{1}, t_{2}⟩ is known for all pairs of symmetric generators since all such pairs are conjugate to such a pair in M′_{1} ≅ 3·McL or McL. In particular, t (the same as the element t_{⋆} of Section 10.1.1) commutes with M_{1} = ⟨t_{01}, N^{0}⟩. Let M ≅ M_{11} × 2 be a subgroup of N such that N_{0} ∩ M ≅ M_{10} × 2 (so that M and N_{0} are not conjugate in N). In fact we have that the M_{11}’s in these two groups intersect in A_{6}—this follows from the monomial action of N on the symmetric generators.

Let M_{2} = ⟨M, t⟩. Then we have ensured that ⟨t⟩ has just 11 images under M and that t has 22 images. Moreover, since these images of t (except those in ⟨t⟩) lie at distance 2 in the bipartite graph, they must commute with t and hence be contained in M_{1} which contains all such symmetric generators except possibly t^{±1}. Let τ ≠ t^{±1} be an image of t under M. Then ⟨τ, (M ∩ N_{0})⟩ ⩽ M_{1} which forces ⟨τ, (M ∩ N_{0})⟩ ≅ 3^{1+4}:(M_{10} × 2) or 3^{4}:(M_{10} × 2), where in the former case the 3^{1+4} denotes an elementary abelian group which when regarded as a module for the M_{10} × 2 is uniserial with a single submodule of dimension 1. We already know that ⟨t, M′⟩ = ⟨t^{M′}, M′⟩ = ⟨t^{M′}⟩:M′, of index (at most) 2 in M_{2}, is perfect. (The fact that ⟨t^{M′}⟩ ∩ M′ = 1 follows from the fact that former group is elementary abelian whereas the latter group is non-abelian simple.) Moreover, we have now shown that ⟨t^{M′}⟩ has order at most 3^{6}.

Now from the Atlas of Brauer Characters (ABC), we know that the minimal 3modular representation of M_{11} has degree 5, thus giving rise to the possibilities 3^{5}:M_{11} or 3^{1+5}:M_{11} for ⟨t, M′⟩. (We rule out the groups 3^{5+1}:M_{11} and 3^{5⊕1}:M_{11} on the grounds that they are not perfect.) Now ⟨t⟩ has just 11 images under M′, so the 5-dimensional representation is the one of type 3A_{11}B_{110} (φ_{2} in the ABC) and not 3A_{55}B_{66} (φ_{3} in the ABC).^{1} But the restriction of of this 5-dimensional representation to M_{10} is uniserial of type 3^{1+4}. In any case, M_{2} can have no group 3^{4}:M_{10} and that forces ⟨τ, (M ∩ N_{0})⟩ ≅ 3^{1+4}:(M_{10} × 2) along with M_{1} ≅ 3·McL:2. Now M_{10} has 3′-index in M_{11}. Therefore, Gaschütz’s Theorem implies that a group of index 11 in 3^{1+5}:M_{11} must have shape 3^{(1⊕1)+4}:M_{10} which contains no subgroup 3^{1+4}:M_{10} and so cannot contain ⟨τ, (M ∩ N_{0})′⟩, a contradiction. Therefore we have ⟨t, M′⟩ ≅ 3^{5}:M_{11} and M_{2} ≅ 3^{5}:(M_{11} × 2) (since the outer involution of M_{2} acts non-trivially on the 3^{5}). Hence ⟨τ, (M ∩ N_{0})⟩ ≅ 3^{1+4}:(M_{10} × 2) contains t which shows that M_{1} also contains t. Let M_{12} = M_{1} ∩ M_{2}. We have already seen that M_{12} = ⟨t, τ, (M ∩ N_{0})⟩ ≅ 3^{1+4}:(M_{10} × 2).

We now focus our attention on M_{3} := ⟨t, t_{10}, H_{10}, t_{01}, t_{i0}, v, σ⟩. (See the 3^{2|4}:2A_{5}.D_{8} paragraph of Section 10.1.2 for a definition of σ.) From our work on 3·McL:2 in Chapter 8, we see that M_{13} = M_{1} ∩ M_{3} ⩾ P = ⟨t, t_{10}, H_{10}, t_{01}, t_{i0}, v⟩ ≅ 3^{2|4}:4S_{5}. Note, however, that σ normalises P (since σ swaps ⟨t⟩ with ⟨t_{10}⟩ and normalises ⟨t_{01}⟩, H_{10} and ⟨v⟩, but that σ ∉ M_{1}, since σ does not normalise ⟨t⟩. Therefore, M_{3} contains P to index 2 and this forces M_{13} to have index 2 in M_{3} and M_{13} = P. From our knowledge of McL, we see that M_{123} = M_{1} ∩ M_{2} ∩ M_{3} has shape 3^{5}:(2 × M_{9}). From our knowledge of the subgroups of N and how M_{2} and M_{3} were defined, we see that M_{23} = M_{2} ∩ M_{3} contains a 2 × M_{9}:2 of N. Now M_{123} has index at most 2 in M_{23}, since M_{13} has index 2 in M_{3}, and so we conclude that this index is exactly 2 and that M_{23} ≅ 3^{5}:(2 × M_{9}:2). It is obvious that M_{3}/O_{3}(M_{3}) has order 960 and a normal 2·A_{5} of index 8. But Aut(2·A_{5}) ≅ S_{5}, so that

1The Atlas of Brauer Characters does not define classes/representations for M_{11} uniquely. However, if x and y are R. A. Wilson’s standard generators for M_{11}, that is o(x) = 2, o(y) = 4, o(xy) = 11 and o(xyxy^{−1}xy^{2}) = 3 (an equivalent definition to his), then we make the convention that xy is in class 11A. All other classes/characters are forced from the 3-modular table.

M_{3}/O_{3}(M_{3}) ≅ [2^{3}].S_{5} (since we already know of a section S_{5}).

Thus we have shown that we have groups M_{1} ≅ 3·McL:2, M_{2} ≅ 3^{5}:(M_{11} × 2) and M_{3} ≅ 3^{2|4}:2A_{5}.D_{8} intersecting in the manner shown in the diagram below.

[figure: 3·McL:2; 3^{1+4}:(M_{10} × 2); 3^{5}:(M_{11} × 2); 3^{2|4}:4S_{5}; 3^{2+2+1}:(2 × 3^{2}:8:_{3}2); 3^{2|4}:2A_{5}.D_{8}; 3^{1+1+2+1}:(2 × 3^{2}:Q_{8})]

Moreover, our groups must generate G as we obviously have t ∈ M_{123} = M_{1} ∩M_{2} ∩M_{3} and N ⩽ ⟨M_{1}, M_{2}⟩. Thus we apply a theorem of Parker and Meierfrankenfeld [29] that tells us that an amalgam of the above type can only generate Ly. (Note that Meierfrankenfeld and Parker’s Theorem was proved under very weak hypotheses. In particular, our weak identification of M_{3} is sufficient.) Since (in principle) we have explicit 111-dimensional matrices over 𝔽_{5} satisfying our presentation, we conclude the following theorem.

Theorem 10.1.1. We have:

\frac{3^{⋆2520}:_{m} 2·A_{11}}{πt_{23}π^{2}t_{23}πt_{23}^{−1}π^{2}t_{23}^{−1} = 1} ≅ Ly,

where π = (1, 3, 9, 5, 4)(2, 6, 7, X, 8) ∈ N^{0}. □

This concludes our section on the Lyons group.

Appendix A

Notation

A.1 Some conventions

Maps usually act on the right. The notation for group structures and algebraic irrationalities is quite close to that in the Atlas. However, we have slightly altered the group structure notation to make it more precise. We have introduced the notation p^{m|n} for special groups and our notation for elementary abelian sections sometimes shows its structure as a module. We give some examples below. The notation for simple groups (and their close relatives) follows that in the Atlas and we give some examples below.

A.2 Notation for particular groups

m or C_{m}  The cyclic group of order m.
ℤ or C_{∞}  The infinite cyclic group.

D_{2m} The dihedral group of order 2m and degree m, and hence has presentation ⟨x, y | x^{m} = y^{2} = 1, x^{y} = x^{−1}⟩.

Q_{4m} The ‘quaternion’ group of order 4m which may be defined by the presentation ⟨x, y | y^{4} = 1, x^{m} = y^{2}, x^{y} = x^{−1}⟩.

p^{m}, E_{p^{m}}  The elementary abelian group of order p^{m}.
p\begin{matrix}1+2m \\ ± \end{matrix}  The extraspecial groups of order p^{1+2m} and ±-type, where p is a prime.
A_{n}  The alternating group on n letters.
S_{n}  The symmetric group on n letters.

Dih(A)

The group with presentation ⟨A, t | t^{2} = 1 = aa^{t}; ∀ a ∈ A⟩, where A is an abelian group.

m:_{a}n  The group ⟨x, y | x^{m} = y^{n} = 1, x^{y} = x^{a}⟩ where a^{n} ≡ 1 (mod m).

F_{n} A Frobenius group of order n. This notation will only be used when the kernel is a p-group. For example, F_{20} means 5:4(≅ 5:_{2}4), F_{21} means 7:3(≅ 7:_{2}3) and F_{pq} means q:p, where p and q are primes and q ≡ 1 (mod p).

L_{n}(q)

The projective special linear groups in n dimensions over 𝔽_{q}. Also denoted by PSL_{n}(q) and A_{n−1}(q) (the latter only for n ⩾ 2). Relatives are SL_{n}(q), GL_{n}(q), PGL_{n}(q), PΣL_{n}(q) and so on.

U_{n}(q)

The projective special linear groups in n dimensions over 𝔽_{q^{2}}. Also denoted by PSU_{n}(q) and ^{2}A_{n−1}(q) (the latter only for n ⩾ 2). Relatives are SU_{n}(q), GU_{n}(q), PGU_{n}(q), PΣU_{n}(q) and so on.

S_{2m}(q)

The projective symplectic group in 2m dimensions over 𝔽_{q}. Also denoted by PSp_{2m}(q) and C_{m}(q). Relatives include Sp_{2m}(q), ΣSp_{2m}(q) and ΣS_{2m}(q).

O_{n}^{ε}(q)

The simple composition factor in the orthogonal group of type ε in n dimensions over 𝔽_{q} (at least if n ⩾ 6). Here ε = 0 if n is odd and ε = ±1 if n is even. This group is also denoted by PΩ_{n}^{ε}(q). We have O_{2m+1}(q) ≅ B_{m}(q), O_{2m}^{+}(q) ≅ D_{m}(q) and O_{2m}^{−}(q) ≅ ^{2}D_{m}(q). Relatives include Ω_{n}^{ε}(q), SO_{n}^{ε}(q) and GO_{n}^{ε}(q). If n is odd then the ε(= 0) is usually omitted.

p^{m|n}  A special group of order p^{m+n} whose centre has order p^{m}.

SD_{2^{m}} The semi-dihedral group of order 2^{m} where m ⩾ 4. It is the group C_{2^{m−1}} :_{a}2 where a = 2^{m−2} − 1.

6_{1}·U_{4}(3):2_{2} See the Atlas [10, pp. 52–59].
L_{n}(q):2_{d}  The group L_{n}(q) extended by a diagonal automorphism of order 2.
L_{n}(q):2_{f}  The group L_{n}(q) extended by a field automorphism of order 2.
L_{n}(q):2_{g}  The group L_{n}(q) extended by a graph automorphism of order 2.

L_{n}(q).2_{d×f} The group L_{n}(q) extended by a (diagonal × field) automorphism where both the diagonal and field automorphisms have order 2.

L_{2}(q^{2}):2_{1} The projective general linear group PGL_{2}(q^{2}) ≅ L_{2}(q^{2}):2_{d} consisting of all linear fractional transformations of 𝔽_{q^{2}} ∪ {∞} if q ⩾ 5 is odd. For q = 3 we have L_{2}(q^{2}):2_{1} ≅ PΣL_{2}(9) ≅ S_{6}.

L_{2}(q^{2}):2_{2} The Atlas name for L_{2}(q^{2}):2_{f} if q ⩾ 5 is odd. For q = 3 we have L_{2}(q^{2}):2_{2} ≅ PGL_{2}(9).

L_{2}(q^{2})·2_{3}  The Atlas name for L_{2}(q^{2})·2_{d×f}, for q ⩾ 3 and odd.

7^{1_{2}⊕1_{3}} :3 The group with presentation ⟨x, y, z | x^{7} = y^{7} = z^{3} = [x, y] = 1, x^{z} = x^{2}, y^{z} = y^{4}⟩.

2^{1+4}:S_{5} The group 2^{5} extended by S_{5} such that when the 2^{5} is regarded as an 𝔽_{2}-module for S_{5}, the submodules have dimensions 0, 1 and 5 only.

2^{1⊕(3_{1}+3_{2})}:L_{3}(2)

As above, but here the submodules are 2^{0}, 2^{1}, 2^{3_{1}}, 2^{3_{1}+3_{2}}, 2^{1⊕3_{1}} and 2^{1⊕(3_{1}+3_{2})}, with 2^{3_{1}} ≅ 2̸^{3_{2}} (as L_{3}(2) modules).

2^{8}:L_{3}(2)

A split extension of 2^{8} by L_{3}(2). The notation does not indicate that L_{3}(2) acts irreducibly on the 2^{8} (but that will probably be the case for this particular example).

A.3 Notation for particular group presentations

These are Coxeter presentations and their variants. See Section 1.5 for a definition of Coxeter diagram groups.

G^{p,q,r} The group ⟨t, u, v | t^{2} = u^{2} = (tu)^{2} = 1 = v^{2} = (tv)^{p} = (uv)^{q} = (tuv)^{r}⟩. It

is independent of the ordering of p, q and r.

(p, q, r)

The group ⟨x, y | x^{p} = y^{q} = (xy)^{r} = 1⟩. Its isomorphism type is independent of the ordering of p, q and r.

⟨p, q, r⟩ The group ⟨x, y, z | x^{p} = y^{q} = z^{r} = xyz⟩. Again, its isomorphism type is independent of the ordering of p, q and r.

(p, q, r; s)

The group ⟨x, y | x^{p} = y^{q} = (xy)^{r} = [x, y]^{s} = 1⟩. Yet again, its isomorphism type is independent of the ordering of p, q and r. However, none of p, q or r can be swapped with s.

(p, q | r, s)

The group ⟨x, y | x^{p} = y^{q} = (xy)^{r} = (xy^{−1})^{s} = 1⟩. Its isomorphism type is unaltered by swapping p with q or r with s.

G^{p,q,r,s,t,u,v} The group ⟨t_{0}, t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}, z | t_{i}^{2} = t_{i}t_{i+1}t_{i+3} = z^{2} = 1 = (t_{i}z)^{a_{i}}⟩,

where (a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}) = (p, q, r, s, t, u, v) and the subscripts are to be taken modulo 7. This presentation is not independent of the ordering of p, q, r, s, t, u and v and our definition differs slightly from the one given in the Atlas.

A.4 Notation for particular sets

𝔸  The field of algebraic numbers.
ℂ  The field of complex numbers.
𝔽_{q}  The unique field with q elements (for q ⩾ 2 finite and a prime power).
ℍ  The division ring of quaternions.
ℕ  The set of natural numbers, that is, the set {0, 1, 2, …}.
ℚ  The field of rational numbers.
ℚ(b5)  The extension of ℚ by r5 = \sqrt{5} or b5 = \frac{1}{2}(−1 + \sqrt{5}).
ℝ  The field of real numbers.
ℤ  The ring of integers.
ℤ[i]  The ring of Gaussian integers.
ℤ[ω]  The ring of Eisenstein integers.
ℤ_{n}  The ring of integers modulo n. This is also denoted by ℤ/nℤ.

A.5 General group theoretic notation

a ∈ S  The element a is a member of the set S.
G, H, …  Arbitrary groups.
gh  The product of (group) elements g and h.
g_{1} … g_{m}  The product of elements g_{1}, …, g_{m}.
g^{−1}  The inverse of the element g.
g^{n}  The n^{th} power of g where n is an integer.
g^{h}  The conjugate of g by h, that is, the element h^{−1}gh.
g^{−h}  The conjugate of g^{−1} by h, that is, the element h^{−1}g^{−1}h = (g^{h})^{−1}.
[g, h]  The commutator of g with h, that is, the element g^{−1}h^{−1}gh = g^{−1}g^{h}.
g^{G}  The conjugacy class of G containing g.
S^{G}  The union of G-conjugacy classes intersecting the set S non-trivially.
Ncl_{G}(S)  The normal closure in G of the set S. It is also denoted by ⟨S^{G}⟩.

g ≈ h The elements g and h have the same order. (The real meaning of this is that we can show that g is conjugate to h or h^{−1}.)

g_{i} ∼ h_{i}

There is an isomorphism ρ : G → H such that ρ(g_{i}) = g_{i}.ρ = h_{i} for all i.
Here it is assumed that ⟨g_{i}⟩ = G and ⟨h_{i}⟩ = H.
G ≅ H  The groups G and H are isomorphic.
H ⩽ G  H is a subgroup of G.
H < G  H is a proper subgroup of G, that is, H ⩽ G and H ≠ G.
S ⊆ G  S is a subset of G.
K ◁ G  K is a normal subgroup of G. This includes the possibility that K = G.

K ∩ H The intersection of subgroups H and K (of G). This notation may also be used for the case when K and H are subsets of a set S.

⟨g, h⟩  The subgroup of G generated by the elements g and h of G.
⟨g_{1}, …, g_{m}⟩ The subgroup of G generated by the elements g_{1}, …, g_{m} of G.
⟨K, H⟩  The subgroup of G generated by the subgroups (or subsets) K and H.

KH The set {kh | k ∈ K, h ∈ H}. If K and H are subgroups (of G) then this set has order |K|.|H|/|K ∩ H|. If K or H is normal (in G) then KH ⩽ G. If K ◁ G and H ◁ G then KH ◁ G.

G/K  The quotient of G by a normal subgroup K of G.
O_{p}(G)  The largest normal p-subgroup of G.

A.B A group G(= A.B) having a normal subgroup isomorphic to A, such that the quotient G/A is a group isomorphic to B.

A:B A case of A.B that is split, that is, G has a subgroup isomorphic to B that has trivial intersection with A. This can be completely specified by stating the action ϕ of B on A by conjugation and A:B is sometimes written as A:^{ϕ}B.

A·B  A case of A.B that is non-split.

A × B The direct product of groups A and B, that is, the group having mutually commuting subgroups A and B with trivial intersection.

A^{n}  The direct product of n copies of A.

A ≀ B The wreath product of groups A and B. It is A^{n}:B, where B is a permutation group on n points.

A ⋆ B  The free product of groups A and B.
A ⋆_{C} B  The free product of groups A and B with amalgamated subgroup C.
A^{⋆n}  The free product of n copies of A.

A ∘ B A central product of groups A and B. This is the direct product of A and B quotiented out by a ‘common’ subgroup that is central in both A and B. This common subgroup is usually appended as a subscript to the circle (as in A ∘_{D} B) and its omission usually implies that the largest value of a common centre should be taken. (Even then, there is no guarantee that the product is unique.)

A.B.C These group products are left associated (i.e., A.B.C ≡ (A.B).C) and imply the existence of a normal subgroup isomorphic to A. This applies if we replace ‘.’ by ‘:’, ‘×’, ‘·’ or ‘∘’. A possible exception is that (A × A):2 may be A ≀ 2.

C_{G}(S)  The centraliser in the group G of the set S.
N_{G}(S)  The normaliser in the group G of the set S.
Z(G)  The centre of the group G, sometimes written as Z or even as C_{G}(G).

Φ(G)

The Frattini subgroup of G, i.e., the intersection of all maximal subgroups of G. It is sometimes written as Φ.

Aut(G)  The automorphism group of G.

Inn(G)

The inner automorphism group of G. It the set the automorphisms induced from conjugation by g for all g ∈ G.

Out(G)  The outer automorphism group of G. It is the group Aut(G)/ Inn(G).

Aut_{B}(C)

The group N_{Aut(B)}(C)/C_{Aut(B)}(C) where C ⩽ B. It is a subgroup of Aut(C).

G′  The derived subgroup of G. So G′ = ⟨[g, h] | g, h ∈ G⟩.
G′′  The derived subgroup of G′. The second derived subgroup of G.
G^{(n)}  The n^{th} derived subgroup of G.

A.6 Notation pertaining to progenitors

T^{⋆m}  The free product of m groups isomorphic to T.
N  The control subgroup (assumed transitive on n points).
t_{i}  The i^{th} symmetric generator (in its more restricted sense).
i  Shorthand for t_{i}.
ī  Shorthand for t_{i}^{−1}.

T_{i} The T_{i} are the n subgroups isomorphic to T which are permuted by N. In most cases we have T_{i} = ⟨t_{i}⟩.

T  The collection {t_{0}, t_{1}, …, t_{n−1}}.
T̄  The collection {T_{0}, T_{1}, …, T_{n−1}}.
N^{i}  The group C_{N}(T_{i}).
N^{ij}  The group C_{N}(⟨T_{i}, T_{j}⟩) = N^{i} ∩ N^{j}.
N^{i_{1}i_{2}...i_{m}}  The group C_{N}(⟨T_{i_{1}}, T_{i_{2}}, …, T_{i_{m}}⟩) = N^{i_{1}} ∩ N^{i_{2}} ∩ … ∩ N^{i_{m}}.
N_{i}  The group N_{N}(T_{i}).
N_{ij}  The group N_{N_{i}}(T_{j}) = N_{i} ∩ N_{j} = N_{N_{j}}(T_{i}).
N_{i_{1}i_{2}...i_{m}}  The group N_{i_{1}} ∩ N_{i_{2}} ∩ … ∩ N_{i_{m}}.
N^{(w)}  The coset stabiliser in N of Nw. It is {π ∈ N | Nwπ = Nw}.
T^{⋆n}:N  A progenitor, usually with permutation action.
T^{⋆n}:_{m} N  A progenitor with (proper) monomial action.

M(T, n)

The full group of ‘monomial automorphisms’ of T^{⋆n}. It is Aut(T) ≀ S_{n} and is often denoted by M.

⋆ The (right N-)coset N. We also use ⋆ to denote the identity element when the more usual notations of 1 and e would cause confusion.

[[⋆]]  The double coset NeN = N.
[[w]]  The double coset NwN.
g ∼ h  The cosets Ng and Nh are equal.

𝔗 The meaning of this depends on the context. It is either the set of symmetric generators and all their powers or just the set of symmetric generators and their inverses.

l(w)

The (shortest possible) length of the coset Nw when written in terms of the elements of the T_{i}.

L(w)

The length of the coset Nw in terms of the symmetric generators and their inverses.

A.7 Atomic algebraic irrationalities (over ℂ)

The notation for atomic algebraic irrationalities follows that in the Atlas [10]. In particular, we write ‘i’ for a square root of −1 and then we have ni = n × i ≠ in = i\sqrt{n}. Also, we have rn = \sqrt{n} and zn = exp(2πi/n) for all n > 1. For n odd (and n > 1) we define bn (and bn^{∗} or bn^{∗∗}) by bn = \frac{1}{2}(−1 + \sqrt{n}), bn^{∗} = \frac{1}{2}(−1 − \sqrt{n}) if n ≡ 1 (mod 4) and bn = \frac{1}{2}(−1 + i\sqrt{n}), bn^{∗∗} = \frac{1}{2}(−1 − i\sqrt{n}) if n ≡ 3 (mod 4). The table below lists all the algebraic irrationalities used in this thesis along with all their algebraic conjugates and their minimal polynomials over ℚ.

b5, b5^{∗}  The Atlas name for the (complex) roots of z^{2} + z − 1 = 0.
b11, b11^{∗∗} The Atlas name for the roots of z^{2} + z + 3 = 0.
i, −i  The Atlas name for the roots of z^{2} + 1 = 0.
r5, −r5  The Atlas name for the roots of z^{2} − 5 = 0.
z3, z3^{∗∗}  The Atlas name for the roots of z^{2} + z + 1 = 0. Also known as ω and ω̄.

A.8 Miscellaneous

⟨χ, ϕ⟩_{G}  The inner product of G-characters χ and ϕ.

ind_{H}^{G} ρ The H-representation/character ρ induced up to G. We shall also denote this by ρ^{G}.

res_{H}^{G} ρ The restriction of the G-representation/character ρ to H. We also denote this by ρ_{H}.

ω A primitive cube root of unity, i.e., ω^{3} = 1 ≠ ω. In fields of characteristic p such that p ≡ 1 (mod 3) a more exacting definition is required (to specify ω up to field automorphism). Namely, if a the first element in (0, 1, 2, 3, …) to have multiplicative order p−1 in 𝔽_{p}, then ω := a^{(p−1)/3}.

ω̄  The ‘conjugate’ of ω, that is, ω̄ = ω^{2} = −1 − ω.
F  An arbitrary field.

F (a)

The (field) extension of F generated by a. Here, a may be transcendental over F (i.e., a is a variable) or a may be algebraic over F.

WLOG  Without loss of generality.
WMA  We may assume.
CFSG  Classification of finite simple groups.
STP  Sufficient to prove.

Appendix B

Some Coxeter groups

In this appendix, we tabulate some of the more interesting Coxeter groups. These will include all the finite ‘classical’ Coxeter groups (with pictures) and some cases of groups of the form G^{p,q,r} and (p, q, r; s). Part of this appendix is devoted to proving the correctness of some basic presentations.

B.1 Proofs of some of the basic presentations

B.1.1 The cases (2, 2, n) ≅ D_{2n} and (2, 3, 2) ≅ D_{6} ≅ S_{3}

It is well-known that a group generated by two involutions is dihedral. For if G = ⟨x, y | x^{2} = y^{2} = (xy)^{n} = 1⟩, then on setting z = xy (so y = xz), we find that G = ⟨x, z | x^{2} = (xz)^{2} = z^{n} = 1⟩, with (xz)^{2} = 1 equivalent to z^{x} = z^{−1}. Finally, (2, 3, 2) is isomorphic to (2, 2, 3) and so is D_{6}.

B.1.2 The case (2, 3, 3) ≅ A_{4} ≅ L_{2}(3)

Let G = ⟨x, y | x^{3} = y^{3} = (xy)^{2} = 1⟩ ≅ (2, 3, 3). Then the image G ≅ A_{4} is realised by setting x ∼ (1, 2, 3) and y ∼ (0, 1, 2). We easily enumerate the 4 cosets in G of H = ⟨x⟩ ≅ C_{3} to get that G has order at most 12—verifying the correctness of the presentation. The 4 cosets are H, Hy, Hyx = Hx^{−1}y^{−1} = Hy^{2} and Hyx^{2} = Hyx^{2}y = Hyyxy = Hyx^{−1}.

B.1.3 The case (2, 3, 4) ≅ S_{4} ≅ PGL_{2}(3)

Let G = ⟨x, y | x^{4} = y^{2} = (xy)^{3} = 1⟩ ≅ (2, 3, 4). Then the image G ≅ S_{4} is realised by setting x ∼ (0, 1, 2, 3) and y ∼ (2, 3). Let H = ⟨u, v⟩ where u = xy and v = y^{x}. Then we have u^{3} = v^{2} = 1 and uv = xyx^{−1}x = x^{−2}.x^{−1}yx^{−1}yx^{−1}.x^{2} = x^{−2}yx^{2} ∼ y so (uv)^{2} = 1 and from Section B.1.1 we get H ≅ S_{3}. We see that the only cosets of H in G are H, Hx, Hx^{2} and Hx^{−1} for we have Hxy = H, Hy = Hx, Hx^{−1}y = Hy^{x}x^{−1} = Hx^{−1} and Hx^{2}y = Hyxy = Hx^{−1}yx^{−1} = Hx^{−2} = Hx^{2}. Thus |G| = 24, proving the correctness of the presentation.

B.1.4 The case (2, 3, 5) ≅ A_{5}

To prove this result we let H = ⟨t, t^{x}⟩, where G = ⟨t, x | t^{2} = x^{3} = (tx)^{5} = 1⟩. Now tt^{x} = txxtx has the same order as xtxtx = txxtxxt = (tx)^{xt} which has the same order as tx, i.e., 5. Thus, H is dihedral of order 10. Coset enumeration of G over H easily yields the 6 cosets: H, Hx, Hxx, Hxt, Hxtx and Hxtxx, thus showing that the order of G is at most 60. The correspondences t ∼ (0, 1)(2, 3), x ∼ (0, 2, 4) and tx ∼ (0, 1, 2, 3, 4) mean that G has A_{5} as a homomorphic image. This yields the result. For the convenience of the reader, I point out that Ht = H, Hxxt = Ht^{x}.xx = Hxx and that Hxtxt = Htxxtxxtxx = Ht.t^{x}.xtxx = Hxtxx.

B.1.5 The case (2, 4, 5; 3) ≅ S_{5}

Let G ≅ ⟨x, y | x^{5} = y^{2} = (xy)^{4} = [x, y]^{3} = 1⟩ ≅ (2, 4, 5; 3). We realise S_{5} as a homomorphic image of G by setting x ∼ (0, 1, 2, 3, 4) and y ∼ (3, 4). Now let u = xy and v = y^{x} and let H = ⟨u, v⟩. Thus we expect H = Stab_{S_{5}}(3) ≅ S_{4}. Moreover, we visibly have u^{4} = v^{2} = 1 and uv = xyx^{−1}yx = x^{2}.x^{−1}yx^{−1}yx^{−1}.x^{2} = x^{2}.yxy.x^{2} ∼ x^{4}.yxy = [x, y] which has order 3. Thus from Section B.1.3 we know that H is a homomorphic image of S_{4}. We enumerate the cosets of H in G to get that H has index 5 in G and hence that G has order at most 120 completing the proof. The cosets of H in G are H, Hx, Hx^{2}, Hx^{−2} and Hx^{−1} on which x acts in the obvious way. Noting that y in an involution, we get Hxy = H since xy ∈ H and hence Hy = Hx, Hx^{−1}y = Hx^{−1}yxx^{−1} = Hx^{−1} since y^{x} ∈ H, Hx^{2}y = Hyxy = Hx[x, y] = Hx[y, x]^{2} = Hxyy^{x}.yy^{x} = Hyy^{x} = Hxx^{−1}yx = Hx^{2} and Hx^{−2}y = Hx^{−1}yx^{−1}y = Hyxyx = Hx^{2}yx = Hx^{−2}.

B.1.6 The case (2, 3, 7; 4) ≅ L_{3}(2)

Here, G is the progenitor 2^{⋆3}:A_{3} factored out by the relations [(0, 1, 2)t_{0}]^{7} = 1 and [t_{0}t_{1}]^{4} = 1 or, equivalently, t_{0}t_{2}t_{1}t_{0}t_{2}t_{1}t_{0} = (0, 2, 1) and t_{0}t_{1}t_{0}t_{1}t_{0}t_{1}t_{0}t_{1} = 1. To prove this, recall from Lemma 3.3.1 (along with Corollary B.4.3) that H = ⟨(0, 1, 2), t_{0}t_{1}t_{0}⟩ ≅ S_{4}. The working below:

\begin{matrix}Ht_{0}t_{1} = & Ht_{0}t_{1}t_{0}.t_{0} = Ht_{0} \\ and Ht_{0}t_{2}t_{1} = & Ht_{0}t_{2}t_{1}t_{0}t_{2}t_{1}t_{0}.t_{0}t_{1}t_{2}t_{0} = H(0, 2, 1).t_{0}t_{1}t_{2}t_{0} = Ht_{0}t_{1}t_{2}t_{0} = \\ Ht_{0}t_{2}t_{0} = & Ht_{0}t_{2}t_{0}t_{2}t_{0}t_{2}t_{0}t_{2}.t_{2}t_{0}t_{2}t_{0}t_{2} = H1.t_{2}t_{0}t_{2}t_{0}t_{2} = Ht_{2}t_{0}t_{2}.t_{0}t_{2} = Ht_{0}t_{2}, \end{matrix}

along with its conjugates under (0, 1, 2), an element of H, implies that H has just 7 cosets in G, namely: H, Ht_{0}, Ht_{1}, Ht_{2}, Ht_{0}t_{2}, Ht_{1}t_{0} and Ht_{2}t_{1}. Thus |G| ⩽ 168. The following permutations on 7 points:

x ∼ (0, 1, 2) ∼ (0, 1, 6)(2, 3, 5)  t ∼ t_{0} ∼ (2, 6)(4, 5)
xt ∼ (0, 1, 2)t_{0} ∼ (0, 1, 2, 3, 4, 5, 6) t^{x} ∼ t_{1} ∼ (0, 3)(2, 4)
[t, x] ∼ t_{0}t_{1} ∼ (0, 3)(2, 6, 4, 5)

realise the isomorphism between G and L_{3}(2).

B.1.7 The case (3, 3 | 4, 5) ≅ 3·A_{6}

Let G ≅ ⟨x, y | x^{3} = y^{3} = (xy)^{4} = (xy^{−1})^{5} = 1⟩ ≅ (3, 3 | 4, 5). Let H = ⟨x, yxy⟩. Now x has order 3, yxy = (xy^{−1})^{y^{−1}} has order 5 and x.yxy = (xy)^{2} has order 2. Therefore, from Section B.1.4, we have H ≅ A_{5}. We establish the order of G by coset enumeration over H. See Example 1.4.2 to see that the image 3·A_{6} is realised.

B.2 The finite Coxeter diagram groups

In this section, the diagrams will be labelled as X_{n} (or X_{n}(m) = I_{2}(m) in the case X = I and n = 2) for X ∈ {A, B, C, D, E, F, G, H, I} and n ⩾ 1 an integer. Such a diagram will have exactly n nodes. For each diagram, we shall give the order and shape of the resulting group. There may be more than one way to exhibit the shape of a group, particularly if it is soluble. The reader we notice that there are no diagrams for X = C or G as we have C_{n} = B_{n} and G_{2} = I_{2}(6). We also have the equalities A_{1} = B_{1}, A_{2} = I_{2}(3), A_{3} = D_{3}, B_{2} = I_{2}(4) and D_{2} = I_{2}(2) = A_{1} × A_{1}. The identity of Weyl(F_{4}) and GO_{4}^{+}(3) does not appear to be well known.

Figure B.1: The finite Coxeter diagram groups

[figure: A_{n}, n ⩾ 1]
[figure: B_{n}, n ⩾ 1; 4]

[figure: D_{n}, n ⩾ 2]

[figure: E_{6}]

[figure: E_{7}]

[figure: E_{8}]
[figure: F_{4}; 4]
[figure: H_{3}; 5]
[figure: H_{4}; 5]
[figure: I_{2}(m), m ⩾ 1; m]

Group  Order of group  Shape of group
A_{n}, n ⩾ 1  (n+1)!  S_{n+1}
B_{n}, n ⩾ 1  2^{n}.n!  2^{n}:S_{n} ≅ 2 ≀ S_{n}
D_{n}, n ⩾ 2  2^{n−1}.n!  2^{n−1}:S_{n}
E_{6}  51 840 = 2^{7}.3^{4}.5  U_{4}(2):2 ≅ S_{4}(3):2
E_{7}  2 903 040 = 2^{10}.3^{4}.5.7  S_{6}(2) × 2
E_{8}  696 729 600 = 2^{14}.3^{5}.5^{2}.7  2·O_{8}^{+}(2):2
F_{4}  1 152 = 2^{7}.3^{2}  GO_{4}^{+}(3)
H_{3}  120 = 2^{3}.3.5  A_{5} × 2
H_{4}  14 400 = 2^{6}.3^{2}.5^{2}  2·(A_{5} × A_{5}):2
I_{2}(m), m ⩾ 1  2m  D_{2m}

B.3 The groups G^{p,q,r}

We give a selection of these. We note that D. F. Holt, W. Plesken and B. Souvignier [23, 24, 34] have effectively proved the infiniteness of G^{3,7,m} when m ⩾ 18 is even (by proving that (2, 3, 7; m) is infinite if m ⩾ 9). We tabulate such groups below. A parameter in brackets is redundant and may be replaced by ∞ or 0.

p, q, r  Shape  p, q, r  Shape
of G^{p,q,r}  of G^{p,q,r}
1, (2), (2) 2^{2} 3, 7, 15 L_{2}(29)
2, 2m, (2m)  D_{4m} × 2 3, 7, 16 2^{6}·PGL_{2}(7)
2, 2m+1, (4m+2) D_{8m+4}  3, 8, 8 PGL_{2}(7) × 2
2, 2m+1, 2m+1 D_{4m+2} 3, 8, 10 3·PGL_{2}(9) × 2
3, 3, (4)  S_{4}  3, 8, 11  PGL_{2}(23)
3, 4, (6)  S_{4} × 2  3, 9, 9  L_{2}(19)
3, 5, (10)  A_{5} × 2 3, 9, 10 L_{2}(19) × S_{3}
3, 5, 5  A_{5}  4, 4, 2m  [16m^{2}]
3, 6, 2m  m^{2}:D_{12} 4, 5, 5  2^{4}·D_{10}
3, 7, 8  PGL_{2}(7) 4, 5, 6  S_{5} × 2
3, 7, 9  L_{2}(8)  4, 5, 8 PGL_{2}(9) × 2
3, 7, 12  PGL_{2}(13) 4, 5, 9  PGL_{2}(19)
3, 7, 13  L_{2}(13)  4, 6, 7 PGL_{2}(13) × 2
3, 7, 14  PGL_{2}(13) 5, 5, 5  L_{2}(11)

B.4 The groups (p, q, r) and (p, q, r; s)

For the most part, we shall concentrate on the case p = 2, q = 3, since then the resulting groups are homomorphic images of the progenitor 2^{⋆3}:A_{3}. In Section 7.3.4, we defined 14 parameters for 2^{⋆3}:A_{3}. Note that 2^{⋆3}:A_{3} quotiented out by parameters a and b is (2, 3, a; b) and that 2^{⋆3}:A_{3} quotiented out by parameter a is (2, 3, a). A selection of images of 2^{⋆3}:A_{3}, including many of the groups (2, 3, a) and (2, 3, a; b), may be found in Table E.1

The following few lemmas provide a generalisation of Lemma 3.3.1 along with some other useful pieces of information. Recall that we are working in the group 2^{⋆3}:A_{3} ≅ 2 ⋆ 3 ≅ ⟨t, x | t^{2} = x^{3} = 1⟩, where x ∼ (0, 1, 2) and t ∼ t_{0}. For notational convenience, we shall let q = tx and r = tx^{2} = tx^{−1}.

Lemma B.4.1. We have q^{−1} = r^{t} and r^{−1} = q^{t}. (Proof exercise.)

Lemma B.4.2. We have that q^{−n−4}r and q^{n}r have the same order for all n ∈ ℤ. Moreover, we have that q^{−2}r has order 2 and that q^{−1}r and q^{−3}r both have order 3.

Proof. We have q^{−n−4}r ∼ q^{n+4}r^{−1} ∼ q^{n}.q^{2}r^{−1}q^{2}. But q^{2}r^{−1}q^{2} = txtxxttxtx = txtxxxtx = txttx = txx = r. So q^{−n−4}r ∼ q^{n}r as required. Also, q^{−1}r = xxttxx = xxxx = x has order 3 and hence so does q^{−3}r. Lastly, q^{−2}r = x^{−1}tq^{−1}r = x^{−1}tx ∼ t of order 2. □

Corollary B.4.3. If q has order 7 then qr and qqr both have the same order. In terms of symmetric generators, we have that if (0, 1, 2)t_{0} has order 7 then t_{0}t_{2} and (0, 1, 2)t_{1}t_{0}t_{2} ∼ t_{2}(0, 1, 2)t_{1}t_{0} = (0, 1, 2)t_{0}t_{1}t_{0} both have the same order.

Lemma B.4.4. If we are in an image of 2^{⋆3}:A_{3} such that a = 7 then we have: a = d_{1} = 7, b = c = f_{2}, d_{2} = e_{2} = f_{4}, e_{1} = 3, e_{3} = f_{3} and f_{1} = 2.

Sketch of proof. Essentially, this is done using identities such as q^{2}r^{−1}q^{2} = r, r^{2}q^{−1}r^{2} = q, q^{−1}r = x and q^{−1}rq^{−1} = t and the like. □

Some of the interesting groups (p, q, r) and (p, q, r; s) that do not occur in the tabulated results for the progenitor 2^{⋆3}:A_{3} in Appendix E are as follows:

(2, 2, n) ≅ D_{2n},
(2, 4, 5; 3) ≅ S_{5},
(2, 5, 6; 2) ≅ S_{5},
(3, 4, 5; 2) ≅ A_{6}
and (2, 5, 9; 2) ≅ L_{2}(19).

We also note that for p an odd prime we have:

⟨s, t | s^{p} = (s^{(p+1)/2}ts^{4}t)^{2} = 1 = (st)^{3} = t^{2}⟩ ≅ L_{2}(p).

Appendix C

The Hoffman–Singleton graph

C.1 Definition

A Hoffman–Singleton graph is a graph of valence 7 such that any two non-adjacent points (vertices) have just one neighbour in common and any two adjacent points have no neighbours in common. Thus there are 7(7 − 1) = 42 points in the graph not joined to a given point. Hence, such a graph must have 1 + 7 + 42 = 50 vertices. A Hoffman–Singleton graph has the following diagram.

[figure: 1; 7; 42; 6; 7; 1; 6; 1]

From the above diagram, it is clear that such a graph has no triangles or quadrangles. Before we give a construction of a Hoffman–Singleton graph, we first say a little about some combinatorial objects associated with S_{6}.

C.2 Some objects associated with S_{6}

Let N ≅ S_{6} act naturally on Ω = {a, b, c, d, e, f}. Then the elements of Ω are called points. The \binom{6}{2} = 15 unordered pairs of distinct elements of Ω are called duads and are written as ab, ac, and so on. A partition of Ω into 3 disjoint pairs is called a syntheme and a typical syntheme is written as ab.cd.ef (for the partition Ω = {a, b} ˙∪{c, d} ˙∪{e, f}). There are just 6!/(3! 2^{3}) = 15 synthemes. Note that this is the same as the number of duads. Furthermore, it is possible to partition the 15 duads into 5 synthemes. We refer to such a partition as a synthematic total or simply total. A typical total is shown below.

ab.cf .de
ac.db.ef
ad.ec.fb
ae.fd .bc
af .be.cd

We use the notation a/bcdef for such a total. There are just 6 totals altogether, the stabiliser of any such being PGL_{2}(5) ≅ S_{5} acting transitively on Ω. The stabiliser of a/bcdef is ⟨(b, c, d, e, f), (a, b)(c, d)(e, f)⟩, which on replacement of abcdef by ∞01234 can be seen to be PGL_{2}(5) in its natural action on projective points.

Of course, the action of N on points induces actions on duads, synthemes and totals. It turns out that the actions of N on points and totals are not isomorphic (the point stabiliser is transitive on totals) but that they are similar (there is an outer automorphism of S_{6} interchanging these two actions). Similarly, the actions of N on duads on synthemes are similar but not isomorphic. The exceptional outer automorphism of S_{6} interchanges points with totals and duads with synthemes. (For n ≠ 6, we have Out(S_{n}) = 1.) The stabilisers in N of a point, duad, syntheme and total are S_{5}, S_{4} ×2, 2≀S_{3} ≅ 2^{3}:S_{3} ≅ S_{4} ×2 and PGL_{2}(5) ≅ S_{5} respectively. The first two of these are intransitive (on points) whereas the latter two are transitive.

C.3 Proof of existence (construction)

Let S = {0, 1, 2, 3, 4, 5, 6} and for i ∈ S let S_{i} = S \ {i}. We define a graph Γ whose vertex set is {⋆} ∪ S ∪ {totals of S_{i} for each i ∈ S} with joins as follows:

(i) ⋆ joins i (for i ∈ S) only.
(ii) For i ∈ S, i is joined to ⋆ and the totals of S_{i} only.

(iii) If T is a total of S_{i} then the joins of T are i along with the six totals T^{(i,j)} (of S_{j}) for i ≠ j.

It is clear that Γ has 50 vertices, each with valence 7. We must now check that there are no triangles in Γ. For x ∈ {⋆} ∪ S it is clear that no two neighbours of x are joined. Let x = T be a total of S_{i}. Then the neighbours of x are i and T^{(i,j)} (i ≠ j). But T^{(i,j)} is a total of S_{j} (and not of S_{i}) and so i and T^{(i,j)} are not adjacent. If T^{(i,j)} and T^{(i,k)} are adjacent, then they are related by T^{(i,k)} = (T^{(i,j)})^{(j,k)} = T^{(i,k,j)} and so T = T^{(i,j)}, a contradiction.

We now show that any two non-adjacent vertices have just one neighbour in common. To fix our notation, we let T and T̄(≠ T) be totals of S_{i} and let T′ ≠ T^{(i,j)} be a total of S_{j}. We have to show that x and y (unequal and non-adjacent) for each of the cases (x, y) = (⋆, T), (j, T), (T, T̄) and (T, T′). It is clear that i is the only vertex adjacent to both ⋆ and T and that T^{(i,j)} is the only vertex adjacent to both j and T. Also, it is easy to see that i is the unique vertex adjacent to T and T̄ for if T^{(i,j)} were adjacent to T̄, since T̄^{(i,j)} is the only total of S_{j} adjacent T̄, we would have to have T^{(i,j)} = T̄^{(i,j)} forcing T = T̄, a contradiction.

We now show that there is a unique common neighbour of T and T′. It is easy to see that this common neighbour, if it exists, is a total of some S_{k}, where k ≠ i or j. Then this common neighbour must be T^{(i,k)} = T^{′(k,j)} and so T′ = T^{(i,j,k)}. Now T^{(i,j,k)} = T^{(i,j)} implies that T^{(j,k)} = T, a contradiction. Also, for i, j, k and l distinct, T^{(i,j,k)} = T^{(i,j,l)} implies that T^{(j,k,l)} = T, also a contradiction. Therefore, the six totals T^{(i,j)} and T^{(i,j,k)} for k ∉ {i, j} are distinct and so must be the six totals of S_{j}. This completes the proof that the graph we have defined is a Hoffman–Singleton graph.

Note that given a total T of S_{i} then for i, j and k distinct the 42 vertices of Γ that are totals of one of the S_{ℓ} have one of the forms: T (1 such), T^{(j,k)} (5 such), T^{(i,j)} (6 such) or T^{(i,j,k)} (30 such). Moreover, each of the totals but T^{(j,k)} has a unique expression as just one of these forms, those of type T^{(j,k)} have three such expressions, all of type T^{(l,m)}. We will not show uniqueness of such a graph in this thesis.

C.4 The automorphism group

Let Γ be the Hoffman–Singleton graph. It is clear from our construction that G = Aut(Γ) contains a subgroup N ≅ S_{7} acting naturally on S, the S_{i}, the totals of the S_{i} and fixing ⋆. Let G_{⋆} = Stab_{Aut(Γ)}(⋆). Then since N ⩽ G_{⋆}, G_{⋆} must act with orbits 1 + 7 + 42 on Γ. (These orbits correspond to those points which are respectively distance 0, 1 or 2 from ⋆ in Γ.) To show that G_{⋆} = N it suffices to show that any automorphism fixing {⋆} ∪ S pointwise is trivial. Let α be an automorphism of Γ that fixes {⋆} ∪ S pointwise. Let T be a total of S_{i}. Then T α is also a total of S_{i}. Then for i, j, k, l, m and n distinct

T —T^{(i,j)}—T^{(i,k,j)}—T^{(i,l,k,j)}—T^{(i,m,l,k,j)}—T^{(i,n,m,l,k,j)}—T^{(n,m,l,k,j)},

is the unique path starting at T and of the form T_{i}—T_{j}—T_{k}—T_{l}—T_{m}—T_{n}—T_{i} where T_{ℓ} denotes a total of S_{ℓ}. Therefore (T α)^{(n,m,l,k,j)} = (T^{(n,m,l,k,j)})α and if we choose j, k, l, m and n so that T = T^{(n,m,l,k,j)} then we must have T α = (T α)^{(n,m,l,k,j)}. But there is just one total of S_{i} that is fixed by (n, m, l, k, j) and so we must have T α = T. Therefore we have shown that α is trivial and that G_{⋆} = N.

Now consider a permutation of the vertices of Γ that:

(i) Fixes 0 and swaps ⋆ with a total T of S_{0},
(ii) Fixes all other totals T̄ of S_{0},
(iii) Swaps i with T^{(0,i)} for all i ∈ S_{0}, and

(iv) Swaps T̄^{(0,i)} with T̄^{(0,k)} for all i ∈ S_{0} and totals T̄ ≠ T of S_{0} where k is the unique element of S \ {0, i} such that T̄ = T^{(i,k)}. (Note that i is the unique element of S \ {0, k} such that T̄ = T^{(k,i)} = T^{(i,k)}, so this is well-defined.)

Call this permutation σ and let 0 < i and 0 < j < k. Then σ, of order 2, can be written in cycle-form as σ = (⋆, T)(0)(i, T^{(0,i)})(T^{(j,k)})(T^{(0,j,k)}, T^{(0,k,j)}) where 0 < i and 0 < j < k. We omit the straightforward, though somewhat tedious, check that σ is an automorphism of Γ. Moreover, ⟨G_{⋆}, σ⟩ is transitive on the vertices of Γ and so

| Aut(Γ)| = |G| = |Γ|.|G_{⋆}| = 50.|S_{7}| = 50.5 040 = 252 000.

In fact, Aut(Γ) ≅ PΣU_{3}(5) ≅ U_{3}(5):2, but we do not show that here.

Appendix D

A computational trick

D.1 Finding the centraliser of an involution

We describe two ways of computationally finding C_{G}(g) where g is an involution and G is a finite group. In both cases we take an element h, which may have to satisfy a further condition, and describe a procedure to obtain an element, k = k(g, h) say, that centralises g. We continue to produce elements k in this manner (by varying h) until we have generators for the whole of C_{G}(g). (We may take g as our first generator for C_{G}(g)). For the first method, h must be an involution, but for the second method, h can be any (non-trivial) element of G.

D.1.1 The first method (the dihedral group method)

For this method, h must be an involution. Then ⟨g, h⟩ is dihedral and C_{⟨g,h⟩}(g) is easily calculated. If gh has odd order then C_{⟨g,h⟩}(g) = ⟨g⟩ and we add no new element to C_{G}(g). However, if gh has even order n = 2m, then C_{⟨g,h⟩}(g) = ⟨g, (gh)^{m}⟩ and we adjoin k = (gh)^{m} to C_{G}(g). Note that since the (gh)^{m} of the previous sentence has order 2, this method can only succeed if C_{G}(g) is generated by involutions. Note also that we need at least three of the (gh)^{m} to generate C_{G}(g) if C_{G}(g)/⟨g⟩ is not dihedral. Also, the words (for the generators of C_{G}(g)) produced by this method tend to be rather long.

We may produce the h as conjugates of a particular involution, x say. It is also better that x and g are not conjugate, since then gh is guaranteed to have even order.

D.1.2 The second method

For the second method, h may be any element of G. We find it convenient to divide the analysis into a number of cases. We have only divided the o(h) ⩽ 2 case off from the other two so that we can avoid (to a certain extent) adding ‘useless’ elements to C_{G}(g). It is unlikely that o(h) ⩽ 2 if h is a ‘random’ element of G.

Case (i): o(h) ⩽ 2

We get no new element to add to C_{G}(g) if h = 1 or gh has odd order. If gh has even order n = 2m we add (gh)^{m} as in the ‘dihedral group method’ above.

Case (ii): o(h) ⩾ 3, [g, h] has even order n = 2m

Since g has order 2, [g, h] = gg^{h}. Thus g commutes with (gg^{h}) = [g, h]^{m}. Also, [g, h^{−1}] has order 2m. We add both [g, h]^{m} and [g, h^{−1}]^{m} to C_{G}(g).

An alternative way to see that g commutes [g, h]^{m} is as follows. The relation [g, h]^{2m} = 1 asserts that [g, h]^{m} = [g, h]^{−m} = [h, g]^{m}. But (since o(g) = 2) [h, g] = [g, h]^{g} and so [g, h]^{m} = ([g, h]^{g})^{m} = ([g, h]^{m})^{g} and so g and [g, h]^{m} commute.

Case (iii): o(h) ⩾ 3, [g, h] has odd order n = 2m + 1

Here we have [g, h]^{m} = [g, h]^{−m−1} = [h, g]^{m+1} = h^{−1}gh[g, h]^{m}g. Thus g.h[g, h]^{m} = h[g, h]^{m}.g and so g commutes with κ(g, h, m) = h[g, h]^{m}. Note also that [g, h^{−1}] has order 2m + 1 and so g commutes with κ(g, h^{−1}, m) = h^{−1}[g, h^{−1}]^{m}. Unfortunately, κ(g, h^{−1}, m) = κ(g, h, m)^{−1} and so we only add h[g, h]^{m} to C_{G}(g).

D.1.2.1 Summary

If [g, h] has order 2m then add [g, h]^{m} and [g, h^{−1}]^{m} to C_{G}(g). If [g, h] has order 2m + 1 then add h[g, h]^{m} to C_{G}(g).

D.1.2.2 Some comments

Since this new method incorporates (in theory at least) the traditional involution centraliser method, it is obviously better. Also, this method can find involution centralisers when they are not generated by involutions—unlike the traditional method.

Lastly, if o([g, h]) = 2, then g commutes with g^{h} = g.[g, h] and g^{h^{−1}} = g.[g, h^{−1}]. If o([g, h]) = 1 then g commutes with h (obviously) if o([g, h]) = 3 then g commutes with hgh^{−1}gh. These comments are useful in 3-transposition groups where g is in the class of 3-transpositions.

D.1.2.3 An example—U_{6}(2)

Robert Wilson [40] defines standard generators for U_{6}(2) to be x and y where x is in class 2A, y has order 7, xy has order 11 and xy^{2} has order 18. The pair (x, y) is unique up to conjugacy in Aut(U_{6}(2)) ≅ U_{6}(2):S_{3} ≅ PΓU_{6}(2). Then x and y satisfy the following presentation:

G = ⟨x, y | x^{2} = y^{7} = (xy)^{11} = 1 = [x, y]^{2} = [x, y^{2}]^{3} = [x, y^{3}]^{3}⟩,

and thus G has U_{6}(2) as a known image. Thus inspection of the presentation reveals that H = ⟨x, x^{y}, x^{y^{−1}}, y^{2}xy^{−2}xy^{2}, y^{3}xy^{−3}xy^{3}⟩ ⩽ C_{G}(x) and considering the known image U_{6}(2) we see that equality holds. (In fact, the generators x and x^{y^{−1}} of C_{G}(x) are redundant.)

Using Magma, we find that H has index 693 in G and the action of G on the cosets of H has image G/K ≅ U_{6}(2) where K = Core_{G}(H) is the kernel of the action. However the action of G on the cosets of H is isomorphic to the action of G on the conjugates of x (since H = C_{G}(x)) and so we find that K = C_{G}(⟨x^{G}⟩). The relations x^{2} = y^{7} = (xy)^{11} = 1 show:

(i) G is perfect, and

(ii) G = ⟨x^{G}⟩.

Thus K is central in G and G ≅ K.U_{6}(2) is a perfect central extension of U_{6}(2). Thus, since the Schur multiplier of U_{6}(2) is 2^{2} × 3, we find that G is a homomorphic image of (2^{2} × 3)·U_{6}(2) and we merely consider all the possible cases to identify G. We exhibit 2^{2}·U_{6}(2) as a homomorphic image of G by taking a coset action of G on the cosets of ⟨xy^{2}xy^{2}, y^{2}xy^{2}x⟩ of degree 2 688. However, 3·U_{6}(2) is not an image of G, since if x̄ and ȳ are preimages of standard generators of U_{6}(2) such that o(x̄) = 2 and o(ȳ) = 7 we must have that o(x̄ȳ) = 33. The calculation in the last sentence was done computationally with 6 × 6 matrices over 𝔽_{4} and we also found that x̄ȳ^{3} has order 11. Therefore, G ≅ 2^{2}·U_{6}(2). (We now also know that ⟨xy^{2}xy^{2}, y^{2}xy^{2}x⟩ ≅ U_{5}(2).)

Appendix E

The tables

E.1 Tabulated results—notation

We now tabulate most of the finite images obtained from this progenitor so far, as well as a couple of infinite ones. In these tables, I have used the convention that boldface entries are the parameters (that is, the corresponding relations) required to define the group and the tiny ones are those implied in the resulting group, G. In the case of s-type parameters, there may be no relation of the required type in our group. In this case, we insert a hyphen (-) in the ‘s’ column. Otherwise we insert the smallest (positive) value of s for which the required relation holds. (There are usually simple criteria to show that there is no s-type relation of a particular sort, for example, in the progenitor 2^{⋆4}:S_{4} if G ≠ ⟨t_{0}, t_{1}, t_{2}⟩ then the parameter s is not realised.)

The other information we have tabulated includes the order and shape of the whole group along with the shapes of subgroups generated by various subsets of the symmetric generators.

The notation for group structure is based on that in the Atlas [10]. We indicate the main differences between our notation and the Atlas notation below. A group that we have labelled as being of shape 3^{2+4}.S does not mean a (special) group of shape 3^{2}.3^{4} extended by S. We use the notation to mean an elementary abelian group of order 3^{6} extended by S and when we regard the 3^{6} as an 𝔽_{3}-module for S it is uniserial with a single submodule of dimension 2. In a group 2^{7}:S, S does not necessarily act irreducibly on the 2^{7}. However 2^{1⊕(3_{1}+3_{2})}:S means that S acts on the 2^{7} with composition factors of dimensions 1, 3 and 3 where the 3-dimensional submodules are non-isomorphic. Moreover, since we use ⊕ for direct sum and + for a uniserial module, S acting on the 2^{7} has just six submodules and these have dimensions 0, 1, 3, 4, 6 and 7. We use 3^{2|4} for a special group whose centre has order 3^{2}. We have sometimes shown the submodule structure of the elementary abelian factors of special and extraspecial groups. We have not attempted to show when an irreducible module is absolutely irreducible. If a group structure is enclosed in square brackets, it indicates a group whose composition factors are those groups within the square brackets. For example, [2^{4}3(A_{5})^{2}] indicates a group with composition factors:

C_{2} four times, C_{3} once and A_{5} twice.

Actually, for modules marked with subscripts, these refer to appropriate representations in the ‘modular Atlas’. For example, in 2^{1⊕(3_{1}+3_{2})}:L_{3}(2), the 3_{1} and 3_{2} refer to modules with characters φ_{2} and φ_{3} in the 2 modular character table of L_{3}(2) given in [27]. On the other hand, in 2^{1⊕(3+3′)}:S, the 3 and 3′ denote arbitrary non-isomorphic 3dimensional 𝔽_{2}-modules of S. A sensible interpretation should be given to such constructs as 7^{1_{2}⊕1_{3}}:3 and 7^{1_{2}⊕1_{2}}:3 even though the ‘modular Atlas’ does not give the 7-modular character table of C_{3}. It is worth noting that non-isomorphic modules can give rise to isomorphic groups, for example we have 2^{8_{2}}:A_{9} ≅ 2^{8_{3}}:A_{9}. (Here 8_{1} is the deleted permutation module of A_{9} and 8_{2} and 8_{3} are the other two irreducible 8-dimensional 𝔽_{2}-modules.)

Finally, we note that all examples of L_{2}(8)^{2}:3 that appear here are of the type where the 3 acts differently on the two L_{2}(8)’s. Our notation does not show this.

E.2 Accuracy of the tabulated results

In tables this extensive, it is probably inevitable that mistakes will arise, though here we believe that they are rather infrequent. We have only found one mistake in the tables so far, and that was of the clerical variety. (We found many more mistakes in the text of [3].) We make no claim that the results are complete in any sense (e.g., we make no claim that we have all images of order less than 10^{6} or that we have all finite images we can get with the parameters we are using). Also, for a progenitor like 2^{⋆3}:A_{3} it is quite obvious that the table is nowhere near complete. Anyway, results from progenitors such as this seem to be closer to (2, 3)-presentations/fingerprints of a particular group rather than a symmetric presentation. Naturally we could be more precise with some of our group structure notation, but this doesn’t count as an error.

E.3 Some sample proofs of the tabulated results

We have provided some such proofs in Section 7.8 and we provide some more below. The examples below are probably harder (and more interesting) the the typical examples.

E.3.1 The images S_{7} and 3·S_{7} of 2^{⋆4}:S_{4}

As ever, the order of G was verified by coset enumeration over N. We easily see that G′ = ⟨N, t_{0}t_{1}⟩ has index 2 in G. Moreover in G′/G′′ we have 1 = [(1, 2), t_{0}t_{1}] = t_{2}t_{0}t_{0}t_{1} = t_{2}t_{1}, so 1 = t_{i}t_{j} for all i and j, thus (2, 3) = (t_{0}t_{1})^{3} = 1 and so (i, j) = 1 for all i and j. Thus G′ is perfect which means that G ≅ (A_{7} × 3):2 for example. We realise S_{7} as a homomorphic image of G by setting x ∼ (2, 6)(0, 5, 4, 3), y ∼ (1, 2)(3, 4) and t ∼ (1, 4)(2, 3)(5, 6).

E.3.2 The image ^{2}F_{4}(2)′ × 2 of 2^{⋆4}:S_{4}

From our work on the progenitor 2^{⋆3}:S_{3} (see Table E.2), we know we have enough relations so that ⟨t_{0}, (0, 1, 2), (0, 1)⟩ is a homomorphic image of L_{3}(3):2. Coset enumeration over this subgroup gives the expected index. The smallest faithful permutation representation is over this subgroup, and working in this representation we find that G ≅ S × 2 where S is a simple group of order 17971200. Using the CFSG we find that S ≅ ^{2}F_{4}(2)′, the Tits group. One wonders if the relation e = 12 is redundant. Certainly ⟨t_{0}, t_{1}, t_{2}⟩ has the correct index without it but it is required to ensure that ⟨t_{0}, t_{1}, t_{2}⟩ ≅ L_{3}(3). This presentation appears in Hammas’s thesis [22]. Also we found that ⟨t_{0}t_{1}, t_{2}t_{3}t_{2}⟩ ≅ L_{3}(3):2 was not conjugate to ⟨t_{0}, (0, 1, 2), (0, 1)⟩, but that their respective derived subgroups were conjugate. We also found that ⟨t_{0}t_{1}, t_{2}t_{3}t_{2}, (0, 1)⟩ ≅ L_{3}(3):2 × 2 was a maximal subgroup of G and did not contain (2, 3).

Now we have better computers, we have been able to determine that e = 12 is redundant by enumerating the (1 497 600) cosets of N in G.

E.3.3 The image 2·(L_{2}(19)^{2}):2^{2} of 2^{⋆4}:S_{4}

From Table E.2, we know that the relations force ⟨t_{0}, t_{1}, t_{2}⟩ to be a homomorphic image of PGL_{2}(19). Coset enumeration over this subgroup verifies the order of G. We can exhibit G very nicely as the orthogonal group GO_{4}^{+}(19) whose order is 2.19^{2}.(19^{2} − 1)^{2} = 93571200. To do this we found the explicit matrices as shown below:

x ∼ [ \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{matrix} ] , y ∼ [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ] and t ∼ [ \begin{matrix}11 & 13 & 13 & 13 \\ 13 & 2 & 3 & 3 \\ 13 & 3 & 2 & 3 \\ 13 & 3 & 3 & 2 \end{matrix} ] .

These matrices can be seen to preserve the quadratic form Q(x) = x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}. Unfortunately, after choosing my quadratic form and my x and y, there are only four possible values for t and none of them is particularly nice. Note that all matrices A ∈ G satisfy AA^{T} = I. We found that ⟨N, t_{0}t_{1}t_{2}t_{3}t_{0}t_{1}⟩ ≅ 2·(A_{5} × A_{5}):2.

E.3.4 The image 2^{1+6}:U_{3}(3) of 3^{⋆4}:S_{4}

The order of G was established by coset enumeration over N. We found that G had a faithful permutation representation over ⟨N, t_{0}^{−1}t_{1}⟩ of degree 72. The shape of G is easily verified to be 2^{1+6}.U_{3}(3) using Cayley. To show that the extension is split I managed to find a subgroup isomorphic to U_{3}(3), namely ⟨(0, 1, 2), t_{0}^{−1}t_{1}t_{3}⟩ = ⟨N^{3}, t_{0}^{−1}t_{1}t_{3}⟩. Incidentally, we found that ⟨N, t_{0}^{−1}t_{1}⟩ ≅ 2^{3_{1}⊕3_{2}} :L_{3}(2), with ⟨(0, 1, 2), t_{0}^{−1}t_{1}⟩ = ⟨(0, 1, 2), (0, 1), t_{0}^{−1}t_{1}⟩ = ⟨N^{3}, t_{0}^{−1}t_{1}⟩ ≅ L_{3}(2) showing that the extension is indeed split. We also note that ⟨t_{0}, t_{1}⟩ ≅ 2^{2}.SL_{2}(3) of order 96 has normal subgroups of orders 1, 2, 4, 8, 32 and 96 only, the centre having order 2 and the derived subgroup having index 3. If P is the Sylow 2-subgroup of ⟨t_{0}, t_{1}⟩ then P′ has order 2 and Z(P) = Φ(P) ≅ 2^{3} has order 8 and P/P′ ≅ 4^{2}.

E.3.5 The image 2^{8_{2}}:A_{9} of 3^{⋆4}:S_{4}

From our work on 3^{⋆3}:S_{3}, we know that ⟨t_{0}, t_{1}, t_{2}⟩ is a homomorphic image of (2^{4_{1}}×2^{4_{2}}):A_{7}. Coset enumeration over this subgroup gives index 72 and hence establishes the order of G. It was found that ⟨N, t_{0}^{−1}t_{1}⟩ has index 256 = 2^{8} in G (the subgroup being isomorphic to A_{9}) and this gives a faithful permutation action for G. This action is primitive and this forces G to have shape 2^{8}:A_{9}. It was found that the A_{9} acted with orbits of sizes 1 + 120 + 135 on the 2^{8} and that the A_{7} acted with orbits of sizes 1 + 120 + 15 + 15 + 105, the A_{9}-orbit of size 135 splitting into A_{7}-orbits of sizes 15 + 15 + 105. We note that the action of A_{9} on the 2^{8} is not the permutation action, for the action has the wrong restriction to A_{7} (and also in the permutation action the A_{9} would have orbits of sizes 1 + 9 + 36 + 84 + 120 on the 2^{8}).

One wonders if the relation b = 14 is redundant. With or without this relation, ⟨N, t_{0}^{−1}t_{1}⟩ has index 256 and ⟨N, t_{0}^{−1}t_{1}t_{0}^{−1}t_{1}t_{0}^{−1}⟩ has index 120960.

E.3.6 The image L_{4}(3):2_{2} of 3^{⋆5}:S_{5}

The order of G was verified by coset enumeration over N. The permutation representation of G over ⟨t_{0}, (0, 1, 2, 3), (0, 1)⟩ is faithful of degree 260. In this representation, it is quite easy to verify most facts about G. We see that the extension is of 2_{2}-type because we have C_{G}(y) = C_{G}((3, 4)) ≅ U_{4}(2):2 × 2 ≅ O_{5}(3) × 2. We found that ⟨N, t_{0}t_{1}t_{0}⟩ = ⟨N, [t_{0}t_{1}t_{2}]^{2}⟩ ≅ U_{4}(2):2 × 2 and that ⟨N, [(0, 1, 2, 3)t_{0}]^{9}⟩ ≅ U_{4}(2) × 2. We also found that ⟨N, t_{0}^{−1}t_{1}⟩ ≅ 2^{4}:S_{5} where the S_{5} acts on the 2^{4} with orbits of sizes 1 + 5 + 10. (Note that G has a faithful permutation representation over ⟨N, t_{0}t_{1}t_{0}⟩ of degree 117.)

Table E.1: Some finite images of the progenitor 2^{⋆3}:A_{3}

Parameters  Shape of Order of
a b c d_{1} d_{2} e_{1} e_{2} e_{3} f_{1} f_{2} f_{3} f_{4} f_{5} f_{6}  G  G
2  3  2  3  1  2  2  2  3  1  3  3  3  3  S_{3}  6
3  2  3  3  2  2  3  3  3  3  1  3  2  2  A_{4}  12
4  3  2  3  2  4  2  4  3  2  3  3  3  3  S_{4}  24
5  5  3  2  3  3  5  2  5  2  3  5  2  2  A_{5}  60
6 n  6  3  n  2  6  6  3  3 \frac{n}{(2,n)} 3  n  n  n^{2}:6  6n^{2}
7 4  4  7  3  3  3  4  2  4  4  3  3  3  L_{2}(7)  168
7 6  6  7  7  3  7  7  2  6  7  7  13 13  L_{2}(13)  1092
7 7  7  7  6  3  6  6  2  7  6  6  13 13  L_{2}(13)  1092
7  8  8  7  6  3  6  8  2  8  8  6 3  6  2^{3}·L_{3}(2)  1344
7  8  8  7  6  3  6  8  2  8  8  6  6 3  2^{3}·L_{3}(2)  1344
7 8  8  7  6  3  6  8  2  8  8  6  6  6 2^{3_{1}⊕3_{2}}·L_{3}(2) 10752
7 9  9  7  9  3  9  7  2  9  7  9 7  7  2^{6}·L_{2}(8)  32256
7 15 15  7  15  3 15 6  2  15  6  15 5 6  J_{1}  175560
7 21 21  7  20  3 20 20 2  21 20 20 4  4  L_{2}(41)  34440
8 4  6  4  7  8  8  6  3  7  3  3  7  7  PGL_{2}(7)  336
8 5 10  5  15  8  8 10 3  15  4  4  12 12 3·PGL_{2}(9) 2160
9 5 10 10  9  5  5  5  9  5  5  9  10 10  L_{2}(19)  3420
9 5 30 30  9  5 15 15 9  15  5  9  10 10 L_{2}(19) × 3 10260
10 4  8  5  15  8  8  8  4  12  5  3  15 15 3·PGL_{2}(9) 2160
11 4 23 11 12 11 12 23 23  6  12  3  12 12  L_{2}(23)  6072
11 5 5  6  6  6  5  3  5  3  2  11  5  5  L_{2}(11)  660
11 5  5  6  6  6  5  3  5  3 2 11  5  5  L_{2}(11)  660

Table E.2: Some finite images of the progenitor 2^{⋆3}:S_{3}

Parameters  Order of Shape of Shape of Shape of
a b c d e  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
2  6  2 2  6  12  D_{6}  D_{6}  S_{3} × 2 ≅ D_{12}
6 2  2 6  2  12  2  2  S_{3} × 2 ≅ D_{12}
3  4  4 3  2  24  2^{2}  2^{2}  S_{4}
4 3  4 2  3  24  D_{6}  G  S_{4}
6  6 2 6  6  36  D_{6}  D_{6}  D_{6} × D_{6}
4  6  4 2  6  48  D_{6}  S_{4}  S_{4} × 2
6 4  4 6  2  48  2^{2}  2^{3}  S_{4} × 2
5 5  3 3  3  60  D_{10}  G  A_{5}
5  5 3 3  3  60  D_{10}  G  A_{5}
5  5  3 3 3  60  D_{10}  G  A_{5}
5 5 3 3  3  60  D_{10}  G  A_{5}
5 5  3 3 3  60  D_{10}  G  A_{5}
5  5 3 3 3  60  D_{10}  G  A_{5}
5  5 3 3 3  60  D_{10}  G  A_{5}
5  5  3 3 3  60  D_{10}  G  A_{5}
8  12 8 2 12  96  D_{12}  2·S_{4}^{+}  4·S_{4}^{+}
12  8  8 12 2  96  D_{8}  [2^{4}]  4·S_{4}^{+}
5  10 6 3  6  120  D_{10}  A_{5}  A_{5} × 2
10 5  6 6  3  120  D_{10}  G  A_{5} × 2
10  10 3 6  6  120  D_{10}  G  A_{5} × 2
7 8  6 4  8  336  D_{8}  L_{2}(7)  PGL_{2}(7)
7  8 6 4  8  336  D_{8}  L_{2}(7)  PGL_{2}(7)
7  8  6 4 8  336  D_{8}  L_{2}(7)  PGL_{2}(7)
8 7  6 8  4  336  D_{14}  G  PGL_{2}(7)
8 7 6 8  4  336  D_{14}  G  PGL_{2}(7)
8 7  6 8 4  336  D_{14}  G  PGL_{2}(7)
8  8 7 6  6  336  D_{8}  G  PGL_{2}(7)
8 8 7 6  6  336  D_{8}  G  PGL_{2}(7)
10 15 6 6 3  360  D_{30}  G  A_{5} × D_{6}
10  15 6 6 3  360  D_{30}  G  A_{5} × D_{6}
15 10 6 3 6  360  D_{10}  A_{5}  A_{5} × D_{6}
15  10 6 3 6  360  D_{10}  A_{5}  A_{5} × D_{6}

Table E.2: Some finite images of the progenitor 2^{⋆3}:S_{3} (ctd)

Parameters  Order of Shape of Shape of Shape of
a b c d e  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
7 9  9  9  7  504  D_{18}  G  L_{2}(8)
7  9 9  9  7  504  D_{18}  G  L_{2}(8)
9 7  9  7  9  504  D_{14}  G  L_{2}(8)
9 7 9  7  9  504  D_{14}  G  L_{2}(8)
12 12 4  6  6  576  D_{12}  S_{4} × 2^{2}  S_{4} × S_{4}
12 12 4  6  6  576  D_{12}  S_{4} × 2^{2}  S_{4} × S_{4}
12  12 4 6 6  576  D_{12}  S_{4} × 2^{2}  S_{4} × S_{4}
12  12 4  6 6  576  D_{12}  S_{4} × 2^{2}  S_{4} × S_{4}
8 8  14  6  6  672  D_{8}  PGL_{2}(7) PGL_{2}(7) × 2
8  14  6  8 4  672  D_{14}  PGL_{2}(7) PGL_{2}(7) × 2
14 8  6 4 8  672  D_{8}  L_{2}(7) × 2 PGL_{2}(7) × 2
8  10 5 10 8  720  D_{10}  G  PGL_{2}(9)
8 10 5 10 8  720  D_{10}  G  PGL_{2}(9)
8  10 5 10 8  720  D_{10}  G  PGL_{2}(9)
10  8 5  8 10  720  D_{8}  G  PGL_{2}(9)
10 8 5  8 10  720  D_{8}  G  PGL_{2}(9)
10  8 5 8 10  720  D_{8}  G  PGL_{2}(9)
7 13  7  13 6  1092  D_{26}  G  L_{2}(13)
7  13 7 13 6  1092  D_{26}  G  L_{2}(13)
13 7  7  6 13 1092  D_{14}  G  L_{2}(13)
13 7 7  6 13 1092  D_{14}  G  L_{2}(13)
24  24 4 12 12 1152  D_{24}  [2^{6}3]  [2^{7}3^{2}]
10 11  12  4 5 1320  D_{22}  G  PGL_{2}(11)
10 11 12 4 5  1320  D_{22}  G  PGL_{2}(11)
10  11  12 4 5 1320  D_{22}  G  PGL_{2}(11)
11 10  12  5 4 1320  D_{10}  L_{2}(11)  PGL_{2}(11)
11 10 12 5 4  1320  D_{10}  L_{2}(11)  PGL_{2}(11)
11  10  12 5 4 1320  D_{10}  L_{2}(11)  PGL_{2}(11)
12 12 5 10 10 1320  D_{12}  G  PGL_{2}(11)
12 12 5 10 10 1320  D_{12}  G  PGL_{2}(11)
10 12 11 12 4 1320  D_{12}  G  PGL_{2}(11)
12 10 11 4 12 1320  D_{10}  G  PGL_{2}(11)

Parameters  Order of Shape of Shape of  Shape of
a b c d e  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
8 10 10 10 8  1440  D_{10}  PGL_{2}(9)  PGL_{2}(9) × 2
10 8 10 8 10 1440  D_{8}  PGL_{2}(9)  PGL_{2}(9) × 2
15 20  12 3 6  1440  D_{20}  A_{5} × 2^{2}  A_{5} × S_{4}
15  20  12 3 6 1440  D_{20}  A_{5} × 2^{2}  A_{5} × S_{4}
20 15 12  6 3 1440  D_{30}  G  A_{5} × S_{4}
20  15  12 6 3 1440  D_{30}  G  A_{5} × S_{4}
8  12  8 4 12 1536  D_{12}  [2^{8}3]  [2^{9}3]
12 8  8  12 4 1536  D_{8}  [2^{8}]  [2^{9}3]
14  24 6 4 24 2016  D_{24}  L_{2}(7) × D_{6} PGL_{2}(7) × D_{6}
24  14 6 24 4 2016  D_{14}  PGL_{2}(7)  PGL_{2}(7) × D_{6}
8  42 6  8 12 2016  D_{42}  (L_{2}(7) × 3):2 (L_{2}(7) × 6):2
42 8 6 12 8  2016  D_{8}  L_{2}(7) × 2  (L_{2}(7) × 6):2
8 10 15 10 8  2160  D_{10}  G  3·PGL_{2}(9)
10 8 15 8 10 2160  D_{8}  G  3·PGL_{2}(9)
7 12 14  6 14 2184  D_{12}  L_{2}(13)  PGL_{2}(13)
7 14 12  7 12 2184  D_{14}  L_{2}(13)  PGL_{2}(13)
12 7  14 14 6  2184  D_{14}  G  PGL_{2}(13)
14 7  12 12 7  2184  D_{14}  G  PGL_{2}(13)
14  14 7 4 4  2184  D_{14}  G  PGL_{2}(13)
14  14 7  4 4 2184  D_{14}  G  PGL_{2}(13)
10 12  22 12 4 2640  D_{12}  PGL_{2}(11) PGL_{2}(11) × 2
12 10 22 4 12 2640  D_{10}  PGL_{2}(11) PGL_{2}(11) × 2
30  40  24 3 12 2880  D_{40}  SL_{2}(5) ∘ Q_{8} SL_{2}(5) ∘ 2·S_{4}^{−}
40  30  24 12 3 2880  D_{30}  G  SL_{2}(5) ∘ 2·S_{4}^{−}
9 9  10 10 10 3420  D_{18}  G  L_{2}(19)
9 10 9 10 5  3420  D_{10}  G  L_{2}(19)
9  10  9  10 5 3420  D_{10}  G  L_{2}(19)
10 9 9  5 10 3420  D_{18}  G  L_{2}(19)
10 9  9 5 10 3420  D_{18}  G  L_{2}(19)
17 17 5 17 17 4080  D_{34}  G  L_{2}(16)
17 17 5 17 17 4080  D_{34}  G  L_{2}(16)
8 10 30 10 8  4320  D_{10}  3·PGL_{2}(9) 3·PGL_{2}(9) × 2

Parameters  Order of Shape of  Shape of  Shape of
a b c d e  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
10 8  30  8  10  4320  D_{8}  3·PGL_{2}(9) 3·PGL_{2}(9) × 2
14  14  14 4 4  4368  D_{14}  PGL_{2}(13)  PGL_{2}(13) × 2
14 14 14 4  4  4368  D_{14}  PGL_{2}(13)  PGL_{2}(13) × 2
14 14  14  4 4  4368  D_{14}  PGL_{2}(13)  PGL_{2}(13) × 2
9  18  16 4  16  4896  D_{18}  L_{2}(17)  PGL_{2}(17)
18 9  16  16 4  4896  D_{18}  G  PGL_{2}(17)
10 12 7  6  10  5040  D_{12}  G  S_{7}
12 10 7  10  6  5040  D_{10}  G  S_{7}
12 16  16  12 4  6144  D_{16}  [2^{10}]  [2^{11}3]
16 12 16 4  12  6144  D_{12}  [2^{10}3]  [2^{11}3]
9  20 6 19 20  6840  D_{20}  L_{2}(19)  PGL_{2}(19)
20 9 6  20 19 6840  D_{18}  G  PGL_{2}(19)
18 20 5  6  20  6840  D_{20}  G  PGL_{2}(19)
18  20 5 6  20  6840  D_{20}  G  PGL_{2}(19)
20 18 5  20  6  6840  D_{18}  G  PGL_{2}(19)
20  18 5  20 6  6840  D_{18}  G  PGL_{2}(19)
13 13 5  12  12  7800  D_{26}  G  L_{2}(25)
13 13 5  12  12  7800  D_{26}  G  L_{2}(25)
10 66  12 4  30  7920  D_{66}  (L_{2}(11) × 3):2 (L_{2}(11) × 6):2
66 10 12  30 4  7920  D_{10}  L_{2}(11) × 2 (L_{2}(11) × 6):2
12 13 6  8  13 11232  D_{26}  G  L_{3}(3):2
13 12 6  13  8 11232  D_{12}  L_{3}(3)  L_{3}(3):2
8 13 12 12  8 11232  D_{26}  G  L_{3}(3):2
13 8 12  8  12 11232  D_{8}  L_{3}(3)  L_{3}(3):2
8 11 24  24  23 12144  D_{22}  G  PGL_{2}(23)
11 8  24  23  24 12144  D_{8}  L_{2}(23)  PGL_{2}(23)
7 15 29  15  15 12180  D_{30}  G  L_{2}(29)
15 7  29  15  15 12180  D_{14}  G  L_{2}(29)
14  29 5 7  15 12180  D_{58}  G  L_{2}(29)
29  14 5  15 7 12180  D_{14}  G  L_{2}(29)
8 13 26 24  13 15600  D_{26}  G  PGL_{2}(25)
13 8 26 13  24 15600  D_{8}  L_{2}(25)  PGL_{2}(25)

Parameters  Order of Shape of  Shape of  Shape of
a b c d e  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
13 28  26 13 4 19656  D_{28}  L_{2}(27)  PGL_{2}(27)
28 13 26 4 13 19656  D_{26}  G  PGL_{2}(27)
9  20 6 57 20 20520  D_{20}  L_{2}(19)  (L_{2}(19) × 3):2
20 9 6 20 57 20520  D_{18}  G  (L_{2}(19) × 3):2
9 10 18 30 10 20520  D_{10}  L_{2}(19)  L_{2}(19) × D_{6}
10 9  18 10 30 20520  D_{18}  G  L_{2}(19) × D_{6}
7 16 12  8 16 21504  D_{16}  2^{3_{1}⊕3_{2}}·L_{2}(7)  2^{6}·PGL_{2}(7)
16 7  12 16 8 21504  D_{14}  G  2^{6}·PGL_{2}(7)
14 58 5 14 30 24360  D_{58}  G  L_{2}(29) × 2
58 14 5 30 14 24360  D_{14}  G  L_{2}(29) × 2
7  28 10 14 30 24360  D_{28}  L_{2}(29)  PGL_{2}(29)
28 7 10 30 14 24360  D_{14}  G  PGL_{2}(29)
8  32  32 5 32 29760  D_{32}  L_{2}(31)  PGL_{2}(31)
32 8  32 32 5 29760  D_{8}  G  PGL_{2}(31)
16  30 6 5 32 29760  D_{30}  L_{2}(31)  PGL_{2}(31)
30  16 6 32 5 29760  D_{16}  G  PGL_{2}(31)
12 30  44 4 12 31680  D_{30}  (L_{2}(11) × A_{4}):2 PGL_{2}(11) × S_{4}
30 12 44 12 4 31680  D_{12}  PGL_{2}(11) × 2^{2} PGL_{2}(11) × S_{4}
7  22 11 11 21 39732  D_{22}  G  L_{2}(43)
22 7 11 21 11 39732  D_{14}  G  L_{2}(43)
18  60 10 4 6 41040  D_{60}  (L_{2}(19) × 3):2 (L_{2}(19) × S_{3}):2
60  18 10 6 4 41040  D_{18}  PGL_{2}(19)  (L_{2}(19) × S_{3}):2
11 44 6 21 14 79464  D_{44}  L_{2}(43)  PGL_{2}(43)
44 11 6 14 21 79464  D_{22}  G  PGL_{2}(43)
15 29 5 29 29 102660  D_{58}  G  L_{2}(59)
29 15 5 29 29 102660  D_{30}  G  L_{2}(59)
9 14 9 14 9 129024  D_{14}  G  2^{8}:L_{2}(8)
14 9 9  9 14 129024  D_{18}  G  2^{8}:L_{2}(8)
52  52 9 4 54 148824  D_{52}  G  PGL_{2}(53)
52  52 9 54 4 148824  D_{52}  G  PGL_{2}(53)
10 19 7 10 11 175560  D_{38}  G  J_{1}
19 10 7 11 10 175560  D_{10}  G  J_{1}

Parameters  Order of Shape of Shape of  Shape of
a  b c d  e  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
7 19 15  19  15  175560  D_{38}  G  J_{1}
19 7 15  15  19  175560  D_{14}  G  J_{1}
35  36 5 9  36  178920  D_{36}  G  L_{2}(71)
36  35 5  36 9  178920  D_{70}  G  L_{2}(71)
11 12  10 6  12  190080  D_{12}  M_{12}  M_{12}:2
12 11  10  12 6  190080  D_{22}  G  M_{12}:2
8 74 9  74  74  388944  D_{74}  G  PGL_{2}(73)
74 8 9  74  74  388944  D_{8}  G  PGL_{2}(73)
8 15 10  8  21  483840  D_{30}  G  2^{2}·L_{3}(4):S_{3}
15 8 10  21  8  483840  D_{8}  2^{2}·L_{3}(4)  2^{2}·L_{3}(4):S_{3}
13  40  78 4  78  492960  D_{40}  L_{2}(79)  L_{2}(79) × 2
40 13  78  78 4  492960  D_{26}  G  L_{2}(79) × 2
12  15 7 5  15  604800  D_{30}  G  HJ = J_{2}
15  12 7  15 5  604800  D_{12}  G  HJ = J_{2}
48  120 5 10  80  950400  D_{120}  G  2·((A_{6} × L_{2}(11)):2_{2})
120  48 5  80 10 950400  D_{48}  G  2·((A_{6} × L_{2}(11)):2_{2})
9 17 8  8  9  1253376  D_{34}  G  2^{1+8}:L_{2}(17)
17 9 8  9  8  1253376  D_{18}  G  2^{1+8}:L_{2}(17)
19  115 5  115  115 6004380  D_{230}  G  L_{2}(229)
115 19 5  115  115 6004380  D_{38}  G  L_{2}(229)
120 60 238 4 7 13651680  D_{60}  G  L_{2}(239) × 2
60  120 238 7 4 13651680  D_{120}  L_{2}(239)  L_{2}(239) × 2
231? 231? 5 11 115? 48985860? D_{462}?  G  L_{2}(461)?
231 231 5 11  115 48985860  D_{462}  G  L_{2}(461)
10 6  10  10  6  300  D_{6}  5^{2}:D_{6}  5^{2}:D_{12}
16 6  16  8  6  768  D_{6}  8^{2}:D_{6}  8^{2}:D_{12}
6 2m 2m  6  \frac{2m}{(2,m)}  12m^{2}  D_{2m}  m^{2}:2  m^{2}:D_{12}
2m 6  2m \frac{2m}{(2,m)}  6  12m^{2}  D_{6}  Depends on m.  m^{2}:D_{12}
6  ∞  ∞  6  ∞  ℵ_{0}  D_{2∞}  Dih(ℤ^{2})  ℤ^{2}:D_{12}
∞ 6  ∞  ∞  6  ℵ_{0}  D_{6}  Dih(ℤ^{2})  ℤ^{2}:D_{12}

Parameters  Order of Shape of Shape of  Shape of
a b c d  G  ⟨t_{0}, t_{1}, t_{2}⟩  ⟨T ⟩  G
6  4  3 2  96  2^{3}  2^{3}  2^{2+1}:A_{4}
4  6  5  5  120  S_{4}  G  S_{5}
6 4  6  4  192  2^{3}  2^{4}  2^{1+2+1}:A_{4}
8  8  7 3  336  G  G  PGL_{2}(7)
8  8 7 3  336  G  G  PGL_{2}(7)
6  8 3  4  384  [2^{5}]  [2^{5}]  [2^{7}3]
11 5  5  6  660  G  G  L_{2}(11)
11  5 5  6  660  G  G  L_{2}(11)
24  8  21 3  1008  PGL_{2}(7) PGL_{2}(7) PGL_{2}(7) × 3
7  7  6  7  1092  G  G  L_{2}(13)
8  6 5  5  1920  4^{2}:D_{6}  G  2^{4}:S_{5}
8 6 5  5  1920  4^{2}:D_{6}  G  2^{4}:S_{5}
33 5  15  6  1980  L_{2}(11)  L_{2}(11)  L_{2}(11) × 3
8 6 10 5  3840  4^{2}:D_{6}  G  2^{1+4}:S_{5}
8 6  10 5  3840  4^{2}:D_{6}  G  2^{1+4}:S_{5}
18  16 4  9  4896  G  G  PGL_{2}(17)
10 12 5  6  5040  G  G  S_{7}
10 12 5 6  5040  G  G  S_{7}
8 6  20  10 7680  4^{2}:D_{6}  G  4·2^{4}·S_{5}
10 6 12 13 15600  5^{2}:D_{6}  G  PΣL_{2}(25)
16  12 5 5 30720 [2^{4}].(4^{2}:D_{6})  G  [2^{4}].(2^{4}:S_{5})
10  8 7  5 80640  PGL_{2}(9)  G  2·L_{3}(4):2_{3}
9 9 6  7 762048  L_{2}(8)^{2}  L_{2}(8)^{2} (L_{2}(8) × L_{2}(8)):3
110  60 5 6 3326400  G  G  S_{7} × L_{2}(11)
6 2m 3m 2m 24m^{3}  Dih(m^{2}) Dih(m^{3})  Dih(m^{3}):A_{4}
6  ∞  ∞  ∞  ℵ_{0}  Dih(ℤ^{2}) Dih(ℤ^{3})  Dih(ℤ^{3}):A_{4}

Parameters  Order of Shape of Shape of  Shape of
a b c d ϵ_{1} ϵ_{2} e f ϕ s  G  ⟨t_{0}, t_{1}, t_{2}⟩  ⟨T ⟩  G
5  4  6  3  6  5  4  2  3 -  120  S_{4}  G  S_{5}
5  4  6 3 6  5  4  2  3 -  120  S_{4}  G  S_{5}
4  6  4  4  4  3  4  6  2 -  192  2^{3}  2^{3}  2^{2+1}:S_{4}
10 4  6  6  6  5  4  2  6 -  240  S_{4}  S_{5}  S_{5} × 2
6  8  8  8 3 7  7  6  6 2  336  G  G  PGL_{2}(7)
6  8  8  8  3  7  7  6  6 2  336  G  G  PGL_{2}(7)
8  6 4 4  4  6  4  6  2 -  384  2^{3}  2^{4}  2^{1+2+1}:S_{4}
8  6  4 4 4  6  4  6  2 -  384  2^{3}  2^{4}  2^{1+2+1}:S_{4}
8  6  8  8  4 3 8  6  4 -  768  [2^{5}]  [2^{5}]  [2^{5}]:S_{4}
8  12  8  8  4  6  8 12 2 -  768  [2^{4}]  2\begin{matrix}1+4 \\ − \end{matrix}  2\begin{matrix}1+(1+2+1) \\ − \end{matrix}:S_{4}
12 11 5 10 10 5 12 5  4 - 1320  L_{2}(11)  L_{2}(11)  PGL_{2}(11)
6  24  8  8  6 21 14 6  6 - 2016  PGL_{2}(7) PGL_{2}(7) PGL_{2}(7) × D_{6}
14 7  7 14 12 6 12 7 12 - 2184  L_{2}(13)  L_{2}(13)  PGL_{2}(13)
12  33 5 10 10 15 12 15 4 - 3960  L_{2}(11)  L_{2}(11) (L_{2}(11) × 3):2
12  10 12 12 5 5 7  6 10 3 5040  G  G  S_{7}
12  10 12 12 5  5 7 6 10 3 5040  G  G  S_{7}
11 11 12 12 11 12 11 11 11 3 6072  G  G  L_{2}(23)
11 11 12 12 11 12 11 11 11 3 6072  G  G  L_{2}(23)
15 10 10 5 10 10 6  6  3 - 7200  A_{5} × 2  G  A_{5} wr 2
10 8 6 6  6 10 8  4  6 - 7680  4^{2}:S_{3}  2^{1+4}:S_{5}  2^{1+4+1}:S_{5}
18  18 16 16 34 4 18 16 16 - 9792  PGL_{2}(17) PGL_{2}(17) PGL_{2}(17) × 2
8  12 12 12 6  8 13 12 12 3 11232  G  G  L_{3}(3):2
8 12 12 12 6  8 13 12 12 3 11232  G  G  L_{3}(3):2
30  10 10 5 10 20 6  6  3 - 14400  A_{5} × 2  G  2·(A_{5} × A_{5}):2
12 10 12 12 5 15 21 6 10 3 15120  G  G  3·S_{7}
12 10 12 12 5 15 21 6 10 3 15120  G  G  3·S_{7}
12  10 12 12 5 15 21 6 10 3 15120  G  G  3·S_{7}
10 8 12 12 6 20 8  4 12 - 15360 (2^{2} × 2)·S_{4} 2·2^{1+4}·S_{5} 2\begin{matrix}1+(1+4+1) \\ − \end{matrix}:S_{5}
13 10  6 6 10 12 10 10 6 - 15600  5^{2}:D_{6}  G  PΣL_{2}(25)
7  30 30 15 10 14 30 4 14 - 24360  G  G  PGL_{2}(29)
24 8  8  8  6 14 7 6  6 - 43008  PGL_{2}(7)  G  2{1+6 \atop +}:PGL_{2}(7)

Parameters  Order of Shape of  Shape of  Shape of
a b c d ϵ_{1} ϵ_{2} e f ϕ s  G  ⟨t_{0}, t_{1}, t_{2}⟩  ⟨T ⟩  G
40  24  24  24 24 30 4 12 12 - 46080  [2^{6}3]  [2^{5}A_{5}]  [2^{6}3A_{5}]
10 10  8  8  10 7 5  8 10 - 80640  PGL_{2}(9)  G  2·L_{3}(4):2_{3}
10 10  8 8 10 7 5  8 10 - 80640  PGL_{2}(9)  G  2·L_{3}(4):2_{3}
10  10  8  8 5 7 10  8 10 - 80640  PGL_{2}(9)  G  2·L_{3}(4):2_{3}
25 8  14 7  6 24  6  8  4 - 117600  PGL_{2}(7)  G  PΣL_{2}(49)
20 10  8  8  10 14 5  8 10 - 161280  PGL_{2}(9)  G  2^{2}·L_{3}(4):2_{3}
20  10  8 8 10 14 5  8 10 - 161280  PGL_{2}(9)  G  2^{2}·L_{3}(4):2_{3}
9 12 10  10 12 9 7 10 6 - 181440  S_{7}  G  A_{9}
10 11 10 10 12 11 12  5  4 - 190080  L_{2}(11)  M_{12}  M_{12}:2
10 11 10 10 12 11 12  5  4 - 190080  L_{2}(11)  M_{12}  M_{12}:2
12  12  10 10 8 10 11 4 12 - 190080 PGL_{2}(11)  G  M_{12}:2
50 8  14  14  6 24 6  8  4 - 235200  PGL_{2}(7) PΣL_{2}(49) PΣL_{2}(49) × 2
50  14  8 8  8 25  6 4 8 - 235200  L_{2}(7) × 2 PΣL_{2}(49) PΣL_{2}(49) × 2
50  14  8  8  8 25 6 4 8 - 235200  L_{2}(7) × 2 PΣL_{2}(49) PΣL_{2}(49) × 2
10 10  8  8 5 21 30  8 10 - 241920 3·PGL_{2}(9)  G  6·L_{3}(4):2_{3}
12 18  6 6 12 27 18 18 6 - 314928  [23^{4}]  [23^{8}]  [2^{4}3^{9}]
9 12 10 10 12 9  14 10 6 - 362880  S_{7}  G  A_{9} × 2
12 12  10 10 8 10 22 4 12 - 380160 PGL_{2}(11)  M_{12}:2  M_{12}:2 × 2
10 10  8  8  10 21 30  8 10 - 483840 3·PGL_{2}(9) 6·L_{3}(4):2_{3} 6·L_{3}(4):2_{3} × 2
8 12 20  20 10 20 31 12 20 5 744000  G  G  L_{3}(5):2
20  8  12  12  6 20 8 4 12 - 983040  [2^{8}3]  [2^{13}A_{5}]  [2^{14}A_{5}]
82 8  10 10 6 20 10 10 8 - 1062720 PGL_{2}(9) L_{2}(81):2_{2} L_{2}(81):2_{2} × 2
10 12  10 10 12 6 12 30 22 - 1140480  M_{12}:2  M_{12}:2  M_{12}:2 × S_{3}
30  40  30  30 60 20 24 12 3 - 1728000 2·(A_{5} × S_{4}^{−})  G  [2^{3}(A_{5})^{3}]
60  10  22 11 10 61 12 4 5 - 1771440 PGL_{2}(11)  G  PΣL_{2}(121)
60 10 22  22 10 61 12 4 10 - 3542880 PGL_{2}(11) L_{2}(121):2_{2} L_{2}(121):2_{2} × 2
8 12 28  28 14 16 48 12 28 7 11261376  G  G  SL_{3}(7):2
14 16 12  12  8 43 48 14 16 3 11327232  G  G  U_{3}(7):2
16 16  6 6 14 48 16  8  6 - 22654464  8^{2}:D_{6}  U_{3}(7):2  U_{3}(7):2 × 2
10 13  8 8 10 12 12 8 12 - 35942400  L_{3}(3)  ^{2}F_{4}(2)′  ^{2}F_{4}(2)′ × 2
18  18  20 20 18 380 10 4 6 - 93571200 PGL_{2}(19)  2·(L_{2}(19)^{2}):2 2·(L_{2}(19)^{2}):2^{2}

Parameters  Order of  Shape of  Shape of  Shape of
a b c d e  G  ⟨t_{0}, t_{1}, t_{2}⟩ ⟨t_{0}, t_{1}, t_{2}, t_{3}⟩  G
6  6  4  6  4  720  S_{4}  S_{5}  S_{6}
6  6 4  6  4  720  S_{4}  S_{5}  S_{6}
10  10  6 4  4  1920  2^{3}  2^{4}  2^{4}:A_{5} × 2
5  10  12  8  8  1920  [2^{4}]  2\begin{matrix}1+4 \\ − \end{matrix}  2\begin{matrix}1+4 \\ − \end{matrix}:A_{5}
10  10 6 6  6  9720  3^{2}:2  3^{3}:2  3^{4}:(A_{5} × 2)
8 8  14  8  6  117600  L_{2}(7) × 2  G  PΣL_{2}(49)
8  8  14 8  6  117600  L_{2}(7) × 2  G  PΣL_{2}(49)
8  8  14  8 6  117600  L_{2}(7) × 2  G  PΣL_{2}(49)
7  19  15  10  19  175560  G  G  J_{1}
12 12 8 6  8  368640  4^{2}:D_{6}  2^{1+4}:S_{5}  2^{4_{1}+1+4_{2}} :S_{6}
13 13 10 6  10  4680000  5^{2}:D_{6}  PΣL_{2}(25)  S_{4}(5)
12 12 8 8  14  19998720  PGL_{2}(7) 2{1+6 \atop +}:PGL_{2}(7)  L_{5}(2):2
12 12? 8 10  30 516984510873600? 3·PGL_{2}(9) L_{4}(4):2_{g}?  L_{5}(4):2_{g}?
10  10 6 2m 2m  120m^{4}  m^{2}:2  m^{3}:2  m^{4}:(A_{5} × 2)
10  10 6  ∞  ∞  ℵ_{0}  Dih(ℤ^{2})  Dih(ℤ^{3})  ℤ^{4}:(A_{5} × 2)

Table E.6: Some finite images of the progenitor 2^{⋆6}:L_{2}(5)

Parameters  Order of Shape of Shape of  Shape of
a b c d e f s  G  ⟨t_{∞}, t_{0}, t_{1}⟩ ⟨t_{∞}, t_{0}, t_{2}⟩  G
4  10 10 12 12 12 3 1320  G  G  PGL_{2}(11)
8  8  10 10  8  8 2 2160  G  G  3·PGL_{2}(9)
9  9 5  9  10 5 - 3420  A_{5}  G  L_{2}(19)
9  9  5  9  10 5 - 3420  A_{5}  G  L_{2}(19)
9 9  5  9  10 5 - 3420  A_{5}  G  L_{2}(19)
9  9  5 9 10 5 - 3420  A_{5}  G  L_{2}(19)
10  10  6  6 4 4 - 3840  2^{3}  2^{3}  2^{1+4+1}:A_{5}
6 6 12 12  6  6 - 15120  3:S_{4}  3:S_{4}  3·S_{7}
7  21  7  41 20 20 5 34440  G  G  L_{2}(41)
19 15 10 15 10 5 - 175560  A_{5} × 2  G  J_{1}
19 15 10 15 10 5 - 175560  A_{5} × 2  G  J_{1}
19  15 10 15 10 5 - 175560  A_{5} × 2  G  J_{1}
8  8 8  8  6 6 - 190080  4^{2}:D_{6}  4^{2}:D_{6}  M_{12} × 2
6 10 8  10  8  8 - 241920 PGL_{2}(7) 3·PGL_{2}(9)  6·L_{3}(4):2_{3}
8  10 10 12 12 12 3 2703360  G  G  2\begin{matrix}1+10 \\ − \end{matrix}:PGL_{2}(11)

Parameters  Order of Shape of Shape of  Shape of
a b c d e f p  G  ⟨t_{0}, t_{1}, t_{2}⟩ ⟨t_{0}, t_{1}, t_{2}, t_{3}⟩  G
4  6  5  4 3  5  4  720  S_{4}  S_{5}  S_{6}
4  6 5  4  3  5  4  720  S_{4}  S_{5}  S_{6}
4  6  10 4  6  5  4  1440  S_{4}  S_{5}  S_{6} × 2
4 6  10  4  6  5  4  1440  S_{4}  S_{5}  S_{6} × 2
6 5  8  6  4  6  4  1920  2^{3}  2^{4}  2^{4}:S_{5}
6  10  8  6 4  6  4  3840  2^{3}  2^{4}  2^{4}:S_{5} × 2
6 10 12  6 6  9  6  19440  3^{2}:2  3^{3}:2  3^{4}:(S_{5} × 2)
6  10  12 6 6  9  6  19440  3^{2}:2  3^{3}:2  3^{4}:(S_{5} × 2)
7  8  25 14  8  25  6  117600 L_{2}(7) × 2  G  PΣL_{2}(49)
12 10 8 12  8  6  8  122880  [2^{6}]  [2^{9}]  [2^{10}]:S_{5}
14 8  50 14  8  25  6  235200 L_{2}(7) × 2 PΣL_{2}(49) PΣL_{2}(49) × 2
8 12 10  8 6  10  8  737280  4^{2}:D_{6}  2^{1+4}:S_{5}  2^{4_{1}+1+4_{2}+1}:S_{6}
10 17 15 10 5  10  6  979200  A_{5} × 2 (A_{5} × A_{5}):2  S_{4}(4)
10 13 26 10 6  12  10 9360000  5^{2}:D_{6}  PΣL_{2}(25)  S_{4}(5):2
8  24  82 8 10 20 10 6886425600 PGL_{2}(9) L_{2}(81):2_{2} S_{4}(9):2_{d} × 2
6 10 4m 6 2m 3m 2m 240m^{4}  m^{2}:2  m^{3}:2  m^{4}:(S_{5} × 2)
6  10 4m 6 2m 3m 2m 240m^{4}  Dih(m^{2}) Dih(m^{3})  m^{4}:(S_{5} × 2)
6 10  ∞  6  ∞  ∞  ∞  ℵ_{0}  Dih(ℤ^{2})  Dih(ℤ^{3})  ℤ^{4}:(S_{5} × 2)
6  10  ∞ 6 ∞  ∞  ∞  ℵ_{0}  ℤ^{2}:2  ℤ^{3}:2  ℤ^{4}:(S_{5} × 2)

Table E.8: Some finite images of the progenitor 2^{⋆7}:L_{3}(2)

Parameters  Order of Shape of Shape of  Shape of
a b c d e f p q s  G  ⟨t_{0}, t_{1}, t_{3}⟩ ⟨t_{0}, t_{1}, t_{2}, t_{5}⟩  G
7 3  6  4  4  4  4  4 -  1344  2^{2}  2^{3}  2^{3}:L_{3}(2)
14  6  6  4  4 4  4  4 -  2688  2^{3}  2^{3}  2 × 2^{3}:L_{3}(2)
6  6  10 12 6  12 4 7 3  5040  D_{6} × 2^{2}  G  S_{7}
12  6  8  8  8  6  8  7 2  12096  [2^{5}]  PGL_{2}(7)  U_{3}(3):2
12  6  8  8  8 6  8  7 2  12096  [2^{5}]  PGL_{2}(7)  U_{3}(3):2
6  6  10 12 6  12 12 21 3  15120  3^{2}:2 × 2^{2}  G  3·S_{7}
14  6  6 4 4  8  4  4 -  21504  2^{3}  2^{4}  2^{1⊕(3_{1}+3_{2})}:L_{3}(2)
8 12 12  6 12 10  4  4 -  40320  S_{4}  S_{5}  S_{8}
7 12 12  8  8  16  8  8 -  86016  [2^{4}]  [2^{7}]  2^{3_{1}}2^{3_{1}+3_{2}} :L_{3}(2)
12 12 10 12 6  12 4 7 3 322560  S_{4} × 2^{2}  S_{7}  2^{6}:S_{7}
12 12 10 12 6 12  4 7 3 322560  S_{4} × 2^{2}  S_{7}  2^{6}:S_{7}
10 12 8  6 12 10 4 8 - 1774080  S_{4}  2^{1+4}:S_{5}  2·M_{22}:2
10 12 8 6 12 10 4 8 - 1774080  S_{4}  2^{1+4}:S_{5}  2·M_{22}:2
10 12 8 6 12 10 12 8 - 5322240  3:S_{4}  2^{1+4}:S_{5}  6·M_{22}:2
15 12 11 12 6 11 4 11 3 244823040 S_{4} × 2^{2}  L_{2}(23)  M_{24}
15 12 11 12 6 11 4 11 3 244823040 S_{4} × 2^{2}  L_{2}(23)  M_{24}
12 ? 10 12 6 12  ? 21 3 3963617280  ?  3·S_{7}  2^{6⊕12}:3·S_{7}

Parameters  Order of Shape of Shape of  Shape of
a b c d e f  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
3  6  3  3  6  2  54  3^{2}  3^{2}  3{1+2 \atop +}:2
5 4  3  2  6  4  120  A_{4}  A_{5}  S_{5}
9  6  3  3  6 2  162  3{1+2 \atop +}  3{1+2 \atop +}  [23^{4}]
4  8  7  4  8  6  336  L_{2}(7)  L_{2}(7)  PGL_{2}(7)
15  12  3 2 6  4  360  A_{4} × 3 A_{5} × 3  S_{5} × 3
15 4  3  6  6  4  360  A_{4}  A_{5}  (A_{5} × 3):2
12  6 2  3  6  4  576  A_{4}  2^{1+2}:A_{4}  [2^{6}3^{2}]
6 5  5  5  5  3  660  A_{5}  G  L_{2}(11)
7  13  7  13 3 7  1092  G  G  L_{2}(13)
5  8 3  4  6  8  1920  4^{2}:3  2^{4}:A_{5}  2^{4}:S_{5}
7 5  5  6  7  7  2520  A_{5}  G  A_{7}
21  39  7  13 3 7  3276  G  G  L_{2}(13) × 3
6  12 3  6 12 4  3456  A_{4} × 3  [2^{6}3^{2}]  [2^{7}3^{3}]
5 8  6  4  6  8  3840  [2^{5}3]  2^{1+4}·A_{5}  2^{1+4}·S_{5}
15 7 4  4 5 7 20160  L_{3}(2)  G  A_{8} ≅ L_{4}(2)
15 7 4  4 5 14 40320  L_{3}(2)  G  2·A_{8}
15 8  4 7 4 6 40320  L_{3}(2)  A_{8}  S_{8}
7 6 4  7  8  6 40320  4^{2}:3  L_{3}(4)  L_{3}(4):2_{3}
7 6  12  7  8  6 120960 3 × 4^{2}:3 3·L_{3}(4)  3·L_{3}(4):2_{3}
15 8  4  21 4 6 120960  L_{3}(2)  A_{8}  (A_{8} × 3):2
5 10 15  5 10 10 124800  U_{4}(3)  U_{4}(3)  U_{4}(3):2
14 5  5  6 14 7 645120  A_{5}  G  (2^{4_{1}} × 2^{4_{2}}):A_{7}
8 6 13 8 12 8 63078912 13^{2}:3  L_{3}(3)^{2}  (L_{3}(3)^{2}):2

Parameters  Order of Shape of Shape of  Shape of
a b c d e f p q s_{1} s_{2}  G  ⟨t_{0}, t_{1}⟩  ⟨t_{0}, t_{1}, t_{2}⟩  G
6 5  4 4 3 2  6  4 - -  720  A_{4}  A_{5}  S_{6}
6  15 12 12 3 2 6  4 - -  2160  A_{4} × 3  A_{5} × 3  S_{6} × 3
13 13  8  8 6 6 13 13 2 3  5616  SL_{2}(3)  G  L_{3}(3)
7  8  12 12 4 4  8  7 3 2 6048  SL_{2}(3)  G  U_{3}(3)
12 15  4 4 3 6  6  4 - -  8640  A_{4}  A_{5}  (A_{6} × A_{4}):2_{1}
12 5  8  8 3 4  6  8 - - 11520  4^{2}:3  2^{4}:A_{5}  2^{4}·S_{6}
21 24 12 12 4 4 24 7 3 2 18144  SL_{2}(3)  G  U_{3}(3) × 3
9 7 10 5 5 6  7  7 - - 181440  A_{5}  A_{7}  A_{9}
12 10 5 10 5 11 12 12 - - 190080  A_{5}  M_{12}  M_{12}:2
7 8 12 12 4 4  8  7 3 - 774144  [2^{4}3]  G  2^{1+6}:U_{3}(3)
8 20 5 10 5 5 10 6 - - 777600  A_{5}  3·(A_{6} × A_{6}) 3·(A_{6} × A_{6}):2
18 6 7 14 9 9 18 14 - - 1524096 L_{2}(8)  L_{2}(8)^{2}  L_{2}(8)^{2}:S_{3}
8 60 5 10 5 15 10 6 - - 2332800  A_{5}  3·(A_{6} × A_{6}) 3·(A_{6} × A_{6}):S_{3}
21 15 14 7 4 4 5 7 - - 9999360 L_{3}(2)  A_{8}  L_{5}(2)
9 14 10 5 5 6 14 7 - - 46448640  A_{5}  2^{4_{1}⊕4_{2}} :A_{7}  2^{8_{2}} :A_{9}

Table E.11: Some finite images of the progenitor 3^{⋆5}:S_{5}

Parameters  Order of Shape of Shape of  Shape of
a b c d e f p q r  G  ⟨t_{0}, t_{1}, t_{2}⟩ ⟨T \ {t_{4}}⟩  G
10 7  6  5  4 3 2 6 4  5040  A_{5}  A_{6}  S_{7}
10  7  6 5 4 3 2 6 4  5040  A_{5}  A_{6}  S_{7}
18 5 12 9  5 3 3 6 6  9720  3^{3}  3^{4}  3^{4}:S_{5}
30 21 6 15 12 3 2 6 4  15120  A_{5} × 3  A_{6} × 3  S_{7} × 3
30 21  6 15 12 3 2 6 4  15120  A_{5} × 3  A_{6} × 3  S_{7} × 3
30 35 12 15 4 3 6 6 4  302400  A_{5}  A_{6}  (A_{7} × A_{5}):2
6  15 12 6 6 4 6 6 8  368640  [2^{6}3]  [2^{8}3]  2^{2|8}:(3 × S_{5})
12 13 18 12 6 2 3 6 4 12130560  2^{1+2}:A_{4} 3^{4}:(2^{1+2}:A_{4}) L_{4}(3):2_{2}
14 11  9 7 5 5 6 7 7 19958400  A_{7}  A_{9}  A_{11}
30 63 21 15 7 4 4 5 7 20158709760 L_{4}(2)  L_{5}(2)  L_{6}(2)

Table E.12: Some finite images of the progenitor 3^{⋆2}:_{m} D_{8}

Parameters  Order of Shape of  Shape of
a b c d e  G  ⟨t_{0}, t_{1}⟩  G
4  6  6  6  4  72  3^{2}  3^{2}:D_{8}
6 5  4  3  4  120  A_{5}  S_{5}
6  5  4 3 4  120  A_{5}  S_{5}
12 6  6  6 12  216  3{1+2 \atop +}  3{1+2 \atop +}:D_{8}
6  10  4  6  4  240  A_{5}  S_{5} × 2
7  8  8  6  3  336  L_{2}(7)  PGL_{2}(7)
7  8  8  6 3  336  L_{2}(7)  PGL_{2}(7)
12  10 4 6  4  480  A_{5}  (A_{5} × 2^{2}):2
10  12 5 10 12 1320  L_{2}(11)  PGL_{2}(11)
28 8  8  6 12 1344  L_{3}(2)  (L_{3}(2) × 4):2
8  24  6 12 8  2304  [2^{5}3^{2}]  [2^{8}3^{2}]
10 12 10 10 12 2640  L_{2}(11) PGL_{2}(11) × 2
9  16 18 16 9  4896  L_{2}(17)  PGL_{2}(17)
11 11 12 12 6  6072  L_{2}(23)  L_{2}(23)
20 9 20 10 18 6840  L_{2}(19)  PGL_{2}(19)
12 10 8  6  8  7680  2^{4}.A_{5}  2^{4+1+1}.S_{5}
10 17 10 17 10 8160  L_{2}(16)  L_{2}(16):2
24  60 24 12 4 8640  [2.3^{2}A_{5}]  [2^{4}3^{2}A_{5}]
28  14 26 4 28 8736  L_{2}(13) (L_{2}(13) × 2^{2}):2
13 12 8 12 8 11232  L_{3}(3)  L_{3}(3):2
12 12 12 12 12 13824  [2^{6}3^{3}]  [2^{9}3^{3}]
10 13 10 12 6 15600  L_{2}(25)  PΣL_{2}(25)
21  42 40 8 5 68880  L_{2}(41)  PGL_{2}(41)
15 10 10 10 10 124800 U_{3}(4)  U_{3}(4):2

Table E.13: Some finite images of the progenitor 3^{⋆3}:_{m} S_{3}

Parameters  Order of Shape of Shape of  Shape of
a b c d e f p q  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
3  3 2  3  3  2  2  3  18  3  3  3^{2}:2
2 3  4  3  3  2  4  3  24  A_{4}  A_{4}  S_{4}
3 2  4  3  3  4  2  3  24  A_{4}  A_{4}  S_{4}
5  5 3  2  2  5  5  5  60  A_{4}  G  A_{5}
3 6  4  3  3 4 2  3  72  A_{4}  A_{4}  3:S_{4}
6 12  8  6  6 2 8  3  144  SL_{2}(3)  SL_{2}(3)  3:GL_{2}(3)
12 6  8  6  6  8 2 6  144  SL_{2}(3)  SL_{2}(3)  3:GL_{2}(3)
4 4 7  4  4  3  3  3  168  G  G  L_{2}(7)
4 4  7  4  4 3 3  3  168  G  G  L_{2}(7)
4  4 7  4  4 3 3  3  168  G  G  L_{2}(7)
4  4  7 4 4 3 3  3  168  G  G  L_{2}(7)
4  4  7  4  4 3 3 3  168  G  G  L_{2}(7)
12 3  8  3  3 4 8  3  288  4^{2}:3  4^{2}:3  ((4^{2}:3) × 3):2
4 4 14  4  4  6  6  3  336  L_{2}(7)  L_{2}(7)  L_{2}(7) × 2
7 4  6  7  7  8  8 2  336  7:3  L_{2}(7)  PGL_{2}(7)
15 15  6 2 6 10 10 5  360  A_{4} × 3 A_{5} × 3  A_{5} × D_{6}
15 15  6  6 2 10 10 15  360  A_{4}  A_{5}  A_{5} × D_{6}
4 5 5  5  5  5  4  3  360  A_{5}  G  A_{6}
5 4 5  5  5  4  5  4  360  A_{5}  G  A_{6}
7  9 9  7  7  7  7 2  504  G  G  L_{2}(8)
12 12 4  3  3  8  8  6  576  A_{4}  2{1+4 \atop +}:3  SL_{2}(3):S_{4}
6 6 5  5  5 11 11 5  660  A_{5}  G  L_{2}(11)
6 5  11  5  5 3 5  5  660  G  G  L_{2}(11)
6  5  11 5 5 3 5  5  660  G  G  L_{2}(11)
6  5  11  5 5 3 5  5  660  G  G  L_{2}(11)
6  5  11  5  5 3 5 5  660  G  G  L_{2}(11)
4 5 10 5  5 10 4  3  720  A_{5}  A_{6}  A_{6} × 2
4 6  8  6  6 4 8  3  768  2^{5}·A_{4}  2^{5}·A_{4}  2^{5}·S_{4}
4 21 6 21 21 8  8  3  1008  7:3 × 3 L_{2}(7) × 3 (L_{2}(7) × 3):2
21 12 6  7 21 8  8 2 1008  7:3 × 3 L_{2}(7) × 3 (L_{2}(7) × 3):2

Table E.13: Some finite images of the progenitor 3^{⋆3}:_{m} S_{3} (ctd)

Parameters  Order of Shape of  Shape of  Shape of
a b c d e f p q  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
4 5 15 5  5  5 12 3  1080  A_{5} × 3  G  3·A_{6}
7  13 7  7  7  6  7 2 1092  G  G  L_{2}(13)
7  6  6  13 13 3 7  7  1092  13:3  G  L_{2}(13)
7  6 6 13 13 3 7  7  1092  13:3  G  L_{2}(13)
4 5  30  5  5 10 12 3  2160  A_{5} × 3  3·A_{6}  3·A_{6} × 2
5  12  10 15 15 4 10 12 2160  A_{5} × 3  A_{6} × 3  A_{6} × D_{6}
7 7  12  7  7 12 4 2 2184  L_{2}(13)  L_{2}(13)  PGL_{2}(13)
17  9 8  8  8 3 9  9  2448  G  G  L_{2}(17)
9  8 9 17 17 3 9  9  2448  G  G  L_{2}(17)
9  8  9  17 17 3 9  9  2448  G  G  L_{2}(17)
6 5 7 4 4  5  7  4  2520  L_{2}(7)  G  A_{7}
6 5 7  4 4 5  7  4  2520  L_{2}(7)  G  A_{7}
9  10 5  5  5  9 19 5  3420  A_{5}  G  L_{2}(19)
7 7 10 6 5 6 6 7 5040  A_{5}  A_{7}  S_{7}
11 12  11 11 11 3 4  4  6072  G  G  L_{2}(23)
11  12  11 11 11 3 4 4  6072  G  G  L_{2}(23)
21 6  14  7 21 14 12 2 6552 L_{2}(13) × 3 L_{2}(13) × 3 (L_{2}(13) × 3):2
15 4  20 15 15 4 10 12 8640  A_{5} × A_{4}  A_{6} × A_{4}  A_{6} × S_{4}
7  29 5  5  5 15 14 14 12180  A_{5}  G  L_{2}(29)
4  15  15 15 15 16 5 3 14880  G  G  L_{2}(31)
6 5  42 4 4 30 42 4 15120 L_{2}(7) × 3  3·A_{7}  3·A_{7} × 2
7 8  12  7  7 16 16 2 21504 2^{3_{1}⊕3_{2}} :7:3 2^{3_{1}⊕3_{2}}·L_{2}(7) 2^{6}·PGL_{2}(7)
20 7  21 21 21 3 20 20 34440  G  G  L_{2}(41)
7  11 11 7  7 21 11 2 39732  G  G  L_{2}(43)
25  25 8 4 4 5 5 25 58800  L_{2}(7)  G  L_{2}(49)
30  9 6 57 57 4 18 15 61560  19:3 × 3 L_{2}(19) × 3 (L_{2}(19) × 3^{2}):2
21  42 10 7 21 30 30 2 73080 L_{2}(29) × 3 L_{2}(29) × 3 (L_{2}(29) × 3):2
15 5  30 42 42 4 28 42 73080 L_{2}(29) × 3 L_{2}(29) × 3 (L_{2}(29) × 3):2
6 6  16  6  6 4 8  6 73728  [2^{9}3]  [2^{12}3]  [2^{13}3^{2}]
13 13 27  9  9 3 27 27 74412  G  G  L_{2}(53)

Table E.13: Some finite images of the progenitor 3^{⋆3}:_{m} S_{3} (ctd)

Parameters  Order of Shape of Shape of  Shape of
a b c d e f p q  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
4 7 12 7  7  8  16  3 86016 2^{3_{1}⊕3_{2}} :7:3 2^{8}:L_{2}(7) 2^{8}:PGL_{2}(7)
29  15 10 29 29 3 15 15 102660  G  G  L_{2}(59)
12 20  77 20 20 3 15 15 110880  G  G  L_{2}(11) × L_{2}(7)
31  31  15 6 6 3 61 61 113460  G  G  L_{2}(61)
21  21 6 3 4 10 10 15 120960  4^{2}:3  L_{3}(4)  L_{3}(4):S_{3}
5 5  30 5 5 26 26  5 124800  U_{3}(4)  U_{3}(4)  U_{3}(4) × 2
4  37 9 37 37 37 37  3 194472  G  G  L_{2}(73)
36 8  63 68 68 3  9  9 411264  G  G  L_{2}(17) × L_{2}(7)
49  49 7 49 49 4 49 49 456288  G  G  L_{2}(97)
6  12  16 12 12 4 16 12 1179648  [2^{13}3]  [2^{16}3]  [2^{17}3^{2}]
17 9 16 8  8 3  9  9 1253376  G  G  2^{1+8}:L_{2}(17)
4 8 16 8  8 7 19 3 1876896 7^{2}:SL_{2}(7)  G  L_{3}(7)
66  60 11 55 55 3 20 20 4007520  G  G  L_{2}(23) × L_{2}(11)
6 13 7  6 4 12 13  6 4245696  L_{2}(7)  G  G_{2}(3)
6 13 7 4 6 12 13  4 4245696  L_{2}(13)  G  G_{2}(3)
4 7 21 7 7? 9 63? 3 16482816?  G  G  L_{3}(8)?

Parameters  Order of Shape of Shape of Shape of
a b c d e f p  G  ⟨t_{0}, t_{1}⟩ ⟨t_{0}, t_{1}, t_{2}⟩  G
4  4  4 3  6  6  3  288  A_{4}  A_{4}  A_{4}:S_{4}
4  4  4  6  6 3  3  288  A_{4}  2^{4}:3  A_{4}:S_{4}
3 5  5  5  4  5  5  360  A_{5}  G  A_{6}
5  5 3  5  5  4  2  360  A_{4}  A_{5}  A_{6}
5 5 3  5  5  4  2  360  A_{4}  A_{5}  A_{6}
6 10 10 5 4  5  5  720  A_{5}  A_{6}  A_{6} × 2
10 10 6 5  5 4  2  720  A_{4}  A_{5}  A_{6} × 2
10 10 6  15  15  12 2 2160  A_{4} × 3  A_{5} × 3  A_{6} × D_{6}
6 20 20 15 4  15 15 8640 A_{5} × A_{4} A_{6} × A_{4}  A_{6} × S_{4}
20 20 6  15  15 4  6  8640  A_{4}  A_{5}  A_{6} × S_{4}
10 10 6 5 5  8  4 11520  4^{2}:3  2^{4}:A_{5}  2^{4+1}:A_{6}
6 10 10 5  8 5  5 11520  A_{5}  2^{4}:A_{6}  2^{4+1}:A_{6}
6 10 10 5 8  5 5 11520  A_{5}  2^{4}:A_{6}  2^{4+1}:A_{6}
12 10 12 5 10 10 5 190080  M_{12}  M_{12}  M_{12}:2
12 10 12 5 10 10 5 190080  M_{12}  M_{12}  M_{12}:2
12 12 10 10 10 5  1 190080  A_{5}  M_{12}  M_{12}:2
15 15 5 20  20 5  5 388000  A_{5}  G  3·(A_{6} × A_{6})
5 15 15 20 5  20 10 388000 A_{5} × A_{4} 3·A_{6} × A_{5} 3·(A_{6} × A_{6})
6 6 8 14  14 8 4 3753792 L_{2}(7)  L_{3}(7)  L_{3}(7) × 2
7 13 6 13 8 6 13 4245696  3{1+2 \atop +}  L_{3}(3)  G_{2}(3)
6 13 7 13 6 8  6 4245696 L_{2}(7)  G  G_{2}(3)
7 6 13 6 8  13  7 4245696  G  G  G_{2}(3)

Table E.15: Some finite images of the progenitor 3^{⋆3}:_{m} S_{4}

Parameters Order of Shape of Shape of Shape of
a b c  G  ⟨t_{0}, t_{1}⟩  ⟨T ⟩  G
4  9  6  648  3^{2}  3^{3}  3^{3}:S_{4}
6 7 5 2520  A_{5}  G  A_{7}
6 7  10 5040  A_{5}  A_{7}  A_{7} × 2
6  21 5 7560  A_{5}  G  3·A_{7}
6 13 10 15600  A_{5}  L_{2}(25) PΣL_{2}(25)

Table E.16: Some finite images of the progenitor 3^{⋆4}:_{m} 2·S_{4}^{+}

Parameters  Order of Shape of Shape of Shape of Shape of
a b c d e  G  ⟨t_{0}, t_{1}⟩ ⟨t_{0}, t_{1}, t_{2}⟩  ⟨T ⟩  G
6 24 13 39 12 33696  L_{3}(3)  L_{3}(3)  L_{3}(3) L_{3}(3) × 3):2
6 8 13 13 12 11232  L_{3}(3)  L_{3}(3)  L_{3}(3)  L_{3}(3):2
22 5 6  6 10 15840  A_{5}  L_{2}(11)  M_{11}  M_{11} × 2
8  6 4 9  6  3888  3^{2}  3^{3}  3^{4}  3^{4}:2·S_{4}^{+}
8  6  4 3 6  432  3^{2}  3^{2}  3^{2}  3^{2}:GL_{2}(3)
11 5  6  6 5 7920  A_{5}  L_{2}(11)  G  M_{11}

Table E.17: Some finite images of the progenitor 3^{⋆6}:_{m} L_{2}(5)

Parameters  Order of Shape of Shape of Shape of Shape of
a b c d e f p q r s  G  ⟨t_{∞}, t_{0}⟩ ⟨t_{∞}, t_{0}, t_{1}⟩ ⟨t_{∞}, t_{0}, t_{2}⟩  G
5 7 29 15 14 15 7 15 29 14 12180  G  G  G  L_{2}(29)
4 16 16 16 8 15 16 5  8 15 14880  G  G  G  L_{2}(31)
19 19 15 15 6 6 11 11 10 5 175560  A_{5}  L_{2}(11)  G  J_{1}
89 11 44 45 4 45 22 22 45 44 352440  G  G  G  L_{2}(89)
9 9 12 12 12 12 15 15 6 4 1451520  A_{4}  2^{3}:A_{4}  A_{5}  S_{6}(2)
12 12 9 9 15 15 12 12 4 6 1451520  A_{4}  2^{3}:A_{4}  A_{5}  S_{6}(2)
60 60 45 45 15 15 12 12 4 6 87091200  A_{4}  2^{3}:A_{4}  A_{5}  S_{6}(2) × A_{5}
45 45 60 60 12 12 15 15 6 4 87091200  A_{4}  2^{3}:A_{4}  A_{5}  S_{6}(2) × A_{5}

Table E.18: Some finite images of the progenitor 3^{⋆5}:_{m} S_{5}

Parameters  Order of Shape of Shape of Shape of  Shape of
a b c d e f p q r  G  ⟨t_{0}, t_{1}⟩ ⟨t_{0}, t_{1}, t_{2}⟩ ⟨T \ {t_{4}}⟩  G
3 3  4  7  5  5  5  4  3  2520  A_{4}  A_{5}  A_{6}  A_{7}
3  3  4  7  5  5  5  4 3  2520  A_{4}  A_{5}  A_{6}  A_{7}
6  6 4 21 10 10 15 12 6 15120  A_{4} × 3 A_{5} × 3  A_{6} × 3  A_{7} × S_{3}
12 12 6  15  8  8 6 6 4 92160  A_{4}  2^{4}:3  2^{6}:3  2^{8}:(3:S_{5})
12 12 6 15 8  8  6  6 4 92160  A_{4}  2^{4}:3  2^{6}:3  2^{8}:(3:S_{5})
4 12 10 8 12 10 11 6 12 190080 A_{4} × 3  M_{12}  M_{12}  M_{12}:2
6  6 12 35 20 20 15 4 6 302400  A_{4}  A_{5}  A_{6}  A_{7} × S_{5}
4 12 10 24 12 10 33 6 12 570240 A_{4} × 3 M_{12} × 3  M_{12} × 3 (M_{12} × 3):2
12 12 6 13 8  8 12 6 4 12130560  A_{4}  2^{1+2}:A_{4} 3^{4}:(2^{1+2}:A_{4}) L_{4}(3) × 2

Appendix F

Some matrix representations

In this appendix, we give explicit matrix generators for some of the representations that we have mentioned in this thesis. In all cases, the generators we have chosen are standard generators as defined by Wilson [40]. We have used b for the irrationality b11.

The representations are given in the order:

(i) 2·A_{11} as 16 × 16 matrices over ℤ[b11].

(ii) 2·HS as 56 × 56 matrices over ℤ.

(iii) HS:2 as 77 × 77 matrices over ℤ.

[ \begin{matrix}· & 1 & · & · & · & · & · & · & · & · & · & · & · & · & · & · \\ −1 & −1 & · & · & · & · & · & · & · & · & · & · & · & · & · & · \\ · & · & · & 1 & · & · & · & · & · & · & · & · & · & · & · & · \\ · & · & −1 & −1 & · & · & · & · & · & · & · & · & · & · & · & · \\ · & · & · & · & · & · & 1 & · & · & · & · & · & · & · & · & · \\ · & · & · & · & · & · & · & · & 1 & · & · & · & · & · & · & · \\ · & · & · & · & −1 & · & −1 & · & · & · & · & · & · & · & · & · \\ · & · & · & · & · & · & · & · & · & · & · & 1 & · & · & · & · \\ · & · & · & · & · & −1 & · & · & −1 & · & · & · & · & · & · & · \\ · & · & · & · & · & · & · & · & · & · & · & · & · & 1 & · & · \\ · & · & · & · & · & · & · & · & · & · & · & · & · & · & 1 & · \\ · & · & · & · & · & · & · & −1 & · & · & · & −1 & · & · & · & · \\ · & · & · & · & · & · & · & · & · & · & · & · & · & · & · & 1 \\ · & · & · & · & · & · & · & · & · & −1 & · & · & · & −1 & · & · \\ · & · & · & · & · & · & · & · & · & · & −1 & · & · & · & −1 & · \\ · & · & · & · & · & · & · & · & · & · & · & · & −1 & · & · & −1 \end{matrix} ]

·  1  · ·  ·  · ·  · ·  ·  · ·  ⋯
·  ·  1 ·  ·  · ·  · ·  ·  · ·  ⋯
·  ·  · ·  1  · ·  · ·  ·  · ·  ⋯
·  ·  · ·  ·  1 ·  · ·  ·  · ·  ⋯
·  ·  · ·  ·  · ·  1 ·  ·  · ·  ⋯
·  ·  · ·  ·  · ·  · ·  1  · ·  ⋯
·  ·  · ·  ·  · ·  · ·  ·  1 ·  ⋯
·  ·  · ·  2  · · −b+1 · b+2 −b −1  ⋯
·  ·  · ·  ·  · ·  · ·  ·  · ·  1 ⋯
·  · −b−2 · −2 −b −1 b+1 ·  ·  · ·  ⋯
·  · −2 · −2  · · −1 · −b  b 1  b 1 · ·
·  ·  · ·  ·  2 · b+1 ·  ·  2 1 b+2 · 1 ·
·  b  1 −1 −b  · ·  · · b+1  · ·  ⋯
· b+2 b+2 · b+1 b−1 1  · 1  · b+1 ·  ⋯
·  2  2 ·  · −2 · −b−1 · −b−1 −b−2 −1 −2b−1 −1 −1 −1
1 −1 b−1 ·  · −b ·  · ·  ·  · · b+1 ⋯

[\begin{matrix}1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1 & . & . & 1 & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ -1 & -1 & -1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & 1 & . & . & . & 1 & . & -1 & . & . & . & . & 1 & -1 & 1 & -1 & 1 & 1 & -1 & -1 & . & 1 & . & -1 & 1 & . & -1 & . & 1 & . & . & 1 & -1 & . & . & . & . & -1 & -1 & 1 & . & 1 & . & 1 & . & -1 & . & . & 1 & . & . & . & 1 & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ -1 & -1 & -1 & -1 & 2 & -1 & -1 & -2 & 1 & -1 & . & . & . & . & 1 & -2 & -1 & -2 & -1 & 1 & 1 & . & -1 & -1 & . & -3 & 1 & 1 & . & 1 & . & -1 & -1 & 1 & . & -1 & 1 & -1 & 1 & 2 & -1 & -1 & -1 & -1 & . & . & 1 & -2 & 1 & -3 & 1 & -3 & -1 & -3 & -1 & -1 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & 1 & . & . & 1 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & . & -2 & . & -1 & -1 & -1 & -1 & . & . & . & 2 & . & 1 & 1 & 1 & -1 & 1 & . & 1 & -1 & -1 & . & -2 & -1 & 1 & . & -1 & . & -1 & . & . & . & -1 & -1 & -1 & . & 1 & . & . & . & . \\ 1 & . & . & 1 & -2 & 1 & 1 & -1 & . & . & -1 & -1 & -1 & . & -1 & . & -2 & . & . & . & . & . & 1 & 2 & -1 & 1 & . & . & . & 1 & . & 1 & . & . & . & . & . & -1 & -1 & . & . & . & 1 & . & . & -1 & . & . & . & 1 & . & 1 & 1 & 1 & . & 1 \\ 1 & . & -1 & . & -1 & 1 & -1 & -1 & 2 & -1 & -1 & . & -1 & . & 1 & -2 & -1 & -1 & -1 & 1 & 2 & . & . & 1 & -1 & -1 & . & 1 & . & 1 & . & . & . & 1 & . & . & . & . & . & 1 & -1 & . & . & . & . & . & 1 & -1 & 1 & -1 & . & -1 & . & -2 & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1 & 2 & 1 & . & 1 & -1 & . & 3 & 3 & 3 & 3 & -1 & . & 1 & . & 2 & 1 & . & -1 & . & -1 & 2 & 1 & -2 & . & 2 & 1 & -1 & . & 1 & -1 & 2 & 1 & 1 & 1 & 1 & . & 2 & -1 & . & 1 & 1 & 2 & 2 & . & 3 & -1 & 2 & -1 & . & -1 & . & -1 & -1 & 3 & -1 \\ . & 1 & 1 & -1 & -1 & . & 1 & . & -2 & . & -2 & -2 & -1 & 3 & -1 & 2 & -1 & 1 & 2 & . & -1 & 2 & 2 & 1 & -3 & . & . & -2 & . & . & 1 & . & 1 & -2 & 1 & 2 & -1 & . & -1 & -2 & 1 & 1 & 3 & 2 & . & -1 & -1 & 2 & . & 3 & . & 1 & . & 3 & . & 1 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & 1 & 1 & -1 & . & . & 1 & -1 & . & . & -1 & . & . & 2 & . & . & -1 & . & . & . & . & 1 & . & . & -2 & -1 & . & . & 1 & 1 & . & . & . & . & . & . & -1 & -1 & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & -1 & . & -1 & . \\ -1 & -2 & . & . & . & -1 & 1 & -2 & -1 & -1 & -1 & -1 & . & 1 & -1 & . & -2 & -1 & 1 & -1 & . & . & 1 & 1 & -2 & -1 & 1 & -1 & 1 & 1 & . & . & -1 & -1 & . & . & 1 & -1 & -1 & . & . & . & . & . & . & -1 & 1 & -1 & 1 & . & . & . & . & 1 & -1 & -1 \\ . & -2 & -1 & 1 & . & . & -1 & -1 & 1 & -1 & . & . & . & . & 1 & -1 & . & -1 & . & . & 1 & -1 & . & . & . & -1 & . & . & . & . & . & . & -1 & . & . & . & 1 & 1 & . & 1 & -1 & 1 & -1 & . & . & . & 1 & -1 & 1 & -1 & . & -1 & . & -1 & . & -1 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & 1 & . & 1 & 1 & -1 & -1 & . & 1 & 2 & 1 & -2 & . & . & 1 & 1 & . & . & . & -2 & -1 & . & 2 & 1 & -1 & 1 & . & . & . & -1 & . & . & -1 & -1 & . & -1 & . & . & . & -1 & -1 & -2 & . & . & . & -1 & -1 & -1 & . & 1 & 1 & . & -1 & . \\ 1 & 2 & 1 & 1 & . & . & . & 3 & 1 & 2 & 3 & . & 1 & . & . & 2 & 2 & 2 & . & . & -1 & 1 & 1 & -2 & 1 & 3 & . & . & . & . & . & 1 & 1 & . & . & 1 & -1 & 2 & -1 & -1 & 1 & 1 & 2 & 1 & . & 2 & -1 & 3 & -2 & 1 & -1 & 2 & . & 1 & 2 & . \\ -1 & . & . & -1 & . & . & 1 & -2 & -2 & -1 & -2 & . & . & . & . & . & -1 & . & 1 & . & . & -1 & -1 & 1 & . & -1 & -1 & . & . & -1 & . & -1 & . & -1 & . & -1 & -1 & -2 & 1 & . & . & -1 & -1 & -1 & . & -1 & . & -1 & . & . & 1 & . & . & 1 & -2 & 1 \\ 1 & . & 1 & 1 & -1 & 1 & 1 & . & . & . & -1 & . & -1 & 1 & -1 & . & -1 & 1 & 1 & . & . & . & 1 & 1 & -1 & . & . & . & . & 1 & . & . & 1 & -1 & . & . & -1 & . & -1 & -1 & 1 & . & 1 & . & 1 & . & . & . & . & 1 & . & . & . & 1 & . & . \\ . & . & . & -1 & . & . & . & . & . & -1 & -1 & . & . & . & . & -1 & . & . & . & . & 1 & . & . & . & -1 & -1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & 1 & . & . & -1 & . & -1 & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . \\ -1 & . & . & -1 & 1 & . & 1 & -2 & -1 & -1 & -2 & . & -1 & 1 & . & . & -1 & . & 1 & 1 & . & . & -1 & 1 & -1 & -2 & . & . & . & . & . & -1 & . & -1 & . & -1 & -1 & -2 & 1 & . & . & -1 & . & -1 & . & -1 & . & -1 & . & . & 1 & -1 & -1 & . & -2 & . \\ -1 & -2 & -1 & . & 1 & . & . & -1 & -1 & -1 & -1 & . & . & . & 1 & -1 & 1 & -1 & 1 & 1 & 1 & . & -1 & . & 1 & -2 & . & . & . & . & 1 & -1 & -1 & -1 & . & . & 1 & . & 1 & . & -1 & . & -2 & -1 & . & . & 1 & -1 & 1 & -1 & 1 & -1 & . & -1 & -1 & -1 \\ 1 & 1 & . & 1 & . & . & -1 & 3 & 2 & 2 & 3 & . & 1 & -1 & 1 & 1 & 3 & . & -1 & . & . & 1 & . & -2 & 2 & 2 & . & . & . & -1 & . & 1 & . & 1 & . & 1 & . & 2 & . & . & . & 1 & . & 1 & -1 & 2 & . & 2 & -1 & . & -1 & 2 & . & . & 2 & -1 \\ -1 & -2 & -2 & . & 1 & 1 & . & -2 & . & -2 & -2 & . & -1 & . & 1 & -2 & -1 & -1 & . & 1 & 1 & -1 & -1 & 1 & . & -3 & . & 1 & . & . & . & -1 & -1 & . & . & -1 & . & -1 & 1 & 1 & -1 & -1 & -2 & -1 & . & -1 & 1 & -2 & 1 & -2 & 1 & -2 & . & -2 & -2 & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ . & . & . & . & 1 & -1 & -2 & 2 & 2 & 1 & 2 & 1 & 1 & -2 & 1 & -1 & 2 & -2 & -2 & . & . & . & -1 & -2 & 2 & -1 & 1 & 1 & . & -1 & . & . & -1 & 2 & . & . & 1 & 1 & 1 & 1 & -1 & . & -2 & . & -1 & 1 & 1 & . & . & -2 & . & -1 & -1 & -2 & 1 & -1 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . \end{matrix}]

164

[\begin{matrix}. & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ -1 & -1 & -1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & -1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & -1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ -1 & -1 & -1 & . & . & . & 1 & -2 & . & . & -1 & . & . & 1 & . & . & -2 & . & . & . & . & . & . & 1 & -1 & -1 & . & . & 1 & 1 & -1 & . & -1 & . & . & . & . & -1 & -1 & 1 & . & . & . & . & . & . & 1 & -2 & . & -1 & . & . & . & . & -1 & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ -1 & -1 & -1 & . & . & . & 1 & -2 & 1 & . & . & . & . & . & . & -1 & -3 & -1 & -1 & . & . & . & . & 1 & -1 & . & 1 & 1 & 1 & 2 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -2 & -1 & 2 & -1 & -1 & . & -1 & . & . & 1 & -2 & . & -2 & . & . & . & -1 & -1 & . \\ . & -1 & . & . & -2 & . & . & -2 & -2 & -2 & -2 & . & . & . & -1 & -1 & -2 & -1 & . & -1 & 1 & -2 & 1 & 1 & -2 & -1 & . & . & 1 & . & . & -1 & -1 & -1 & . & . & 1 & -1 & -1 & . & . & . & -1 & . & . & -2 & 1 & -1 & 2 & . & . & . & . & 1 & -1 & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & -1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & 1 & . & . & 1 & -1 & 1 & . & 1 & 2 & 1 & -1 & . & 1 & -1 & 1 & -1 & 1 & . & . & -1 & 1 & . & . & . & 1 & . & -1 & . & 1 & -1 & 1 & 1 & . & 1 & . & . & . & -1 & . & 1 & . & 2 & 1 & . & 1 & -1 & . & -1 & . & . & . & . & . & 1 & . \\ . & . & 1 & 1 & . & -1 & -1 & 2 & 1 & 1 & 3 & 2 & 1 & -2 & . & . & 1 & 1 & -1 & -1 & . & -1 & . & -1 & 2 & 4 & . & 1 & -1 & 1 & . & 1 & 1 & 1 & -1 & -1 & 2 & . & . & . & . & . & . & -2 & 1 & 1 & -1 & . & . & 1 & . & . & . & -1 & 2 & -1 \\ . & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & -1 & . & . & . & -1 & -1 & . & . & -1 & . & 1 & . & -2 & -1 & -1 & . & -1 & -1 & -1 & 1 & -2 & . & . & . & . & . & . & -1 & . & . & . & . & . & . & -1 & 2 & . & 1 & . & . & . & -2 & -1 & 1 & -1 & . & -1 & 2 & . & . & -2 & . & -2 & 1 & -1 \\ 1 & . & . & . & . & . & -1 & 1 & . & . & . & . & . & -1 & . & . & 1 & . & . & . & . & . & . & . & 1 & 1 & . & . & -1 & -1 & 1 & . & 1 & . & . & . & . & 1 & 1 & -1 & . & . & . & . & . & . & -1 & 1 & . & 1 & . & . & . & . & 1 & . \\ 1 & . & . & . & -2 & 1 & 1 & -2 & 1 & -1 & -2 & 1 & -1 & -1 & -2 & -1 & -5 & . & -1 & -1 & . & -2 & 1 & 4 & -2 & 3 & . & 1 & -1 & 2 & -1 & 2 & 2 & 1 & -1 & -3 & . & -3 & . & . & . & -2 & 1 & -2 & 1 & -2 & -1 & -2 & . & 2 & . & -1 & . & . & . & 2 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & -1 & 1 & . & . & 1 & -1 & 2 & 1 & . & . & -1 & . & . & . & -1 & -1 & -1 & 1 & 1 & -1 & . & 1 & . & -1 & . & . & . & 1 & -1 & . & . & . & 1 & -1 & . & . & . & 1 & . & . & -1 & . & . & 1 & 1 & -2 & . & -2 & . & -1 & . & -2 & . & -1 \\ -1 & . & -1 & . & . & . & 1 & -2 & . & . & -1 & . & . & 1 & . & . & -2 & . & . & . & . & . & . & 1 & -2 & -1 & . & . & 1 & 1 & -1 & . & -1 & . & . & . & . & -1 & -1 & 1 & . & . & . & . & . & . & 1 & -1 & . & -1 & . & . & . & . & -1 & . \\ . & -1 & . & . & -2 & . & 1 & -2 & -1 & -1 & -3 & -1 & -1 & 1 & -1 & . & -3 & . & 1 & -1 & . & . & 1 & 3 & -3 & . & . & -1 & . & 1 & . & . & . & -1 & . & . & . & -1 & -1 & . & . & . & 1 & . & . & -2 & . & -1 & 1 & 2 & . & . & . & 2 & -1 & 1 \\ . & -1 & -1 & . & . & . & 1 & -3 & . & -1 & -1 & . & . & . & -1 & -1 & -3 & -1 & -1 & -1 & . & -1 & . & 1 & -2 & -1 & . & . & 1 & 1 & -2 & . & -1 & . & . & -1 & 1 & -2 & -1 & 1 & . & -1 & -1 & . & . & -1 & 1 & -3 & 1 & -2 & . & -1 & . & -1 & -1 & . \\ -1 & -1 & . & 1 & 1 & -1 & . & 1 & 1 & 1 & 2 & 1 & . & -1 & -1 & 1 & . & 1 & . & -1 & -1 & -1 & . & . & 1 & 3 & . & . & -1 & 1 & -1 & 1 & 1 & . & . & -1 & 2 & . & . & . & 1 & . & . & -1 & 1 & 1 & -1 & -1 & . & 1 & . & -1 & . & -1 & 2 & -1 \\ . & 2 & 2 & -2 & . & -1 & . & 1 & -3 & . & . & -1 & . & 2 & . & 1 & 2 & 1 & 1 & . & -1 & 2 & . & -2 & -1 & -1 & . & -1 & . & -1 & 1 & -1 & . & -1 & . & 2 & -1 & 1 & . & -1 & 1 & 1 & 2 & 2 & . & . & -1 & 3 & . & 1 & . & 1 & . & 2 & . & 1 \\ . & . & -1 & . & 1 & 1 & . & 1 & 2 & . & -1 & -1 & -1 & 2 & 1 & . & 1 & 1 & 1 & 1 & . & 2 & . & . & -1 & -1 & . & . & . & . & . & 1 & 1 & . & . & 1 & -2 & 1 & 1 & . & . & 1 & 1 & 1 & 1 & 1 & . & 1 & -1 & 1 & . & . & . & -1 & 1 & . \\ -1 & . & 2 & -1 & . & -2 & . & 1 & -3 & . & 1 & . & 1 & . & . & 1 & 2 & . & 1 & -1 & . & . & . & -2 & 1 & -1 & . & -1 & . & -1 & 1 & -1 & -1 & -1 & . & 1 & 1 & 1 & . & -1 & 1 & 1 & -1 & . & . & . & . & 1 & 1 & . & . & . & . & 1 & . & -1 \\ . & 1 & 1 & -1 & -2 & . & 2 & -1 & -3 & . & -2 & -1 & . & 2 & -1 & 2 & -2 & 2 & 2 & -1 & -1 & 1 & 1 & 1 & -2 & 1 & . & -1 & 1 & . & . & . & . & -1 & . & 1 & -1 & -1 & -2 & -1 & 1 & . & 3 & 1 & . & -1 & -1 & 1 & . & 2 & . & 2 & . & 4 & -1 & 2 \\ . & . & -1 & -1 & 1 & 1 & . & -1 & . & -1 & -1 & -1 & . & 1 & 1 & -1 & . & -1 & . & 1 & . & 1 & . & -1 & -1 & -3 & . & . & 1 & . & . & -1 & -1 & . & . & 1 & -1 & . & . & 1 & -1 & . & . & 1 & . & . & 1 & . & . & -2 & . & . & . & -1 & -1 & . \\ . & 1 & . & . & 2 & . & . & 1 & . & 1 & 1 & . & . & . & 1 & 1 & 2 & 1 & 1 & 1 & -1 & 1 & -1 & -1 & 2 & . & . & . & -1 & -1 & 1 & -1 & 1 & . & . & . & -1 & 1 & 1 & -1 & . & . & . & . & . & 1 & -1 & 1 & -1 & . & 1 & . & . & . & . & . \\ 1 & . & . & 1 & -1 & . & . & . & 1 & . & 1 & 2 & . & -3 & -1 & . & -1 & 1 & -1 & -1 & . & -2 & . & 1 & 1 & 4 & . & 1 & -1 & 1 & -1 & 1 & 1 & 1 & -1 & -2 & 1 & -1 & . & . & . & -1 & . & -2 & 1 & . & -1 & -1 & . & 1 & . & . & . & -1 & 1 & . \\ . & -1 & -1 & . & . & . & . & -2 & -1 & -1 & -1 & -1 & . & . & . & -1 & -2 & -1 & . & . & . & . & . & 1 & . & -2 & . & . & 1 & . & . & -1 & -1 & . & . & . & . & -1 & -1 & 1 & -1 & -1 & . & . & -1 & -1 & 1 & -1 & . & -2 & . & . & 1 & 1 & -2 & 1 \\ . & 1 & . & . & . & . & . & 1 & 1 & 1 & . & -1 & -1 & 1 & -1 & 1 & -1 & 1 & . & . & -1 & 1 & 1 & 1 & -1 & 1 & . & . & -1 & . & . & 1 & 2 & . & 1 & . & -1 & . & . & -1 & 1 & . & 2 & 1 & . & . & -1 & 1 & -1 & 2 & . & . & . & 1 & 1 & 1 \\ 1 & 1 & . & . & . & 1 & -1 & 3 & 2 & 1 & 1 & . & . & -1 & 1 & . & 2 & 1 & . & 1 & . & 1 & . & . & 2 & 2 & . & 1 & -1 & -1 & 1 & 1 & 2 & 1 & . & . & -2 & 1 & 1 & -1 & . & . & 1 & . & . & 1 & -1 & 2 & -2 & 2 & . & 1 & . & . & 1 & 1 \\ . & 1 & 1 & -1 & . & . & . & 2 & -1 & . & 1 & 1 & 1 & . & 1 & 1 & 3 & 1 & 1 & . & . & 1 & -1 & -2 & 1 & 1 & . & . & . & -1 & 1 & . & . & . & -1 & 1 & . & 1 & 1 & -1 & . & 1 & . & . & 1 & 1 & -1 & 2 & . & 1 & . & 1 & . & . & 1 & . \end{matrix}]

165

Bibliography

[1] Michael Aschbacher, Finite Group Theory, Cambridge studies in advanced mathematics 10, Cambridge University Press, (1986).

[2] Wieb Bosma and John Cannon, Handbook of Magma Functions, Sydney (March 16, 1994).

[3] John N. Bray, Symmetric Presentations of Finite Groups, M. Phil. (Sc., Qual.) thesis, Birmingham (1995).

[4] John N. Bray, Robert T. Curtis and Ahmed M. A. Hammas, A systematic approach to symmetric presentations II: Generators of order 3, in preparation.

[5] Colin M. Campbell, Symmetric presentations and linear groups, Contemp. Math. 45 (1985), 33–39.

[6] John J. Cannon, An introduction to the group theory language, Cayley. In Computational Group Theory (ed. M. D. Atkinson), Academic Press (1984), pp. 145–183.

[7] John Cannon and Wieb Bosma, Handbook of Cayley functions, Sydney (April 1992).

[8] Roger W. Carter, Simple Groups of Lie Type, John Wiley and Sons, London (1972).

[9] Michael J. Collins, Representations and characters of finite groups, Cambridge studies in advanced mathematics 22, Cambridge University Press, (1990).

[10] John H. Conway, Robert T. Curtis, Simon P. Norton, Richard A. Parker and Robert A. Wilson, An 𝔸𝕋𝕃𝔸𝕊 of Finite Groups, Oxford University Press (1985).

[11] Harold S. M. Coxeter and William O. J. Moser, Generators and relations for discrete groups, 4^{th} Ed., Springer-Verlag (1984).

[12] Robert T. Curtis, The Steiner system S(5, 6, 12), the Mathieu group M_{12} and the “Kitten”. In Computational Group Theory (ed. M. D. Atkinson), Academic Press (1984), pp. 353–358.

[13] Robert T. Curtis, Symmetric presentations I: Introduction, with particular reference to the Mathieu groups M_{12} and M_{24}. In Proceedings of the L.M.S. Durham Conference on ‘Groups and Combinatorics’, (1990), pp. 380–396.

[14] Robert T. Curtis, Geometric interpretation of the ‘natural’ generators of the Mathieu groups, Math. Proc. Cam. Phil. Soc. 107 (1990), 19–26.

[15] Robert T. Curtis, Symmetric presentations II: The Janko group J_{1}, J. London Math. Soc. (2) 47 (1993), 294–308.

[16] Robert T. Curtis, Symmetric generation of the Higman–Sims group, J. Algebra 171 (1995), 567–586.

[17] Robert T. Curtis, Monomial Modular Representations and Construction of the Held group, J. Algebra 184 (1996), 1205–1227.

[18] Robert T. Curtis, Some interesting symmetric presentations, with particular reference to the Suzuki chain and sporadic simple groups, Various notebooks and manuscripts, (unpublished as of September 1997).

[19] Robert T. Curtis, Ahmed M. A. Hammas and John N. Bray, A systematic approach to symmetric presentations I: Involutory generators, Math. Proc. Cam. Phil. Soc. 119 (1996), 23–34.

[20] Leonard E. Dickson, Linear Groups, with an Exposition of the Galois Field Theory, Leipzig (1901), reprinted Dover (1958).

[21] Marshall Hall Jr., The Theory of Groups, Macmillan (1959).

[22] Ahmed M. A. Hammas, Symmetric presentations of some finite groups, Ph. D. thesis, Birmingham (1991).

[23] Derek F. Holt and Wilhelm Plesken, A cohomological criterion for a finitely presented group to be infinite, J. London Math. Soc. (2) 45 (1992), 469–480.

[24] Derek F. Holt, Wilhelm Plesken and Bernd Souvignier, Constructing a representation of the group (2, 3, 7; 11), preprint (1994).

[25] James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics 29, Cambridge University Press, (1990).

[26] Abdul Jabbar, Symmetric presentations of subgroups of the Conway groups and related topics, Ph. D. thesis, Birmingham (1992).

[27] Christoph H. Jansen, Klaus Lux, Richard A. Parker, Robert A. Wilson, An 𝔸𝕋𝕃𝔸𝕊 of Brauer characters, Oxford University Press, (1995).

[28] David L. Johnson, Presentations of Groups, London Math. Soc. Lecture Notes 22, Cambridge University Press, (1976).

[29] Ulrich Meierfrankenfeld and Christopher W. Parker, A computer-free construction of the Lyons Group, in preparation.

[30] Simon P. Norton, F and other simple groups, Ph. D. thesis, Cambridge (1975).

[31] Simon P. Norton, The construction of J_{4}. In Proceedings of the Santa Cruz Group Theory Conference (eds. Cooperstein and Mason), Amer. Math. Soc., (1980), pp. 271–278.

[32] Richard A. Parker, An Integral MeatAxe. In The Atlas 10 years on (eds. R. T. Curtis and R. A. Wilson), pp. 215–228. LMS Lecture Note Series, Cambridge University Press, to appear.

[33] Richard A. Parker and Robert A. Wilson, The computer construction of matrix representations of finite groups over finite fields, J. Symbolic Computation 9 (1990), 583–590.

[34] Wilhelm Plesken and Bernd Souvignier, Analyzing finitely presented groups by constructing representations, preprint (1994).

[35] Edmund F. Robertson and Colin M. Campbell, Symmetric presentations. In Proceedings of the 1987 Singapore conference, Walter de Gruyter (1989), pp. 497–506.

[36] Martin Schönert et al., GAP: Groups, Algorithms and Programming, version 3.4, Lehrstuhl D für Mathematik, RWTH Aachen (10 July, 1994).

[37] Jean-Pierre Serre, Trees, Springer-Verlag, New York, (1980).

[38] Robert A. Wilson, Maximal subgroups of automorphism groups of simple groups, J. London Math. Soc. (2) 32 (1985), 460–466.

[39] Robert A. Wilson, Standard generators for sporadic simple groups, J. Algebra 184 (1996), 505–515.

[40] Robert A. Wilson et al., A Computerised Library of Group Representations, Presently (September 1997) available on the Internet at http://for.mat.bham.ac.uk/atlas/ (still being updated, 1997).