Monomial modular representations and symmetric generation of the Harada–Norton group^{∗}

John N. Bray and Robert T. Curtis, School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT.

       E-mail: jnb@maths.bham.ac.uk, R.T.Curtis@bham.ac.uk.

March 14, 2005

Abstract

This paper is a sequel to Curtis [7], where the Held group was constructed using a 7-modular monomial representation of 3·A_{7}, the exceptional triple cover of the alternating group A_{7}. In this paper, a 5-modular monomial representation of 2·HS:2, a double cover of the automorphism group of the Higman–Sims group, is used to build an infinite semi-direct product P which has HN, the Harada– Norton group, as a ‘natural’ image. This approach assists us in constructing a 133-dimensional representation of HN over ℚ(\sqrt{5}), which is the smallest degree of a ‘true’ characteristic 0 representation of P. Thus an investigation of the low degree representations of P produces HN. As in the Held case, extension to the automorphism group of HN follows easily.

Keywords: sporadic group, symmetric presentation, modular representation, matrix group construction.

1 Introduction

In 1973, Bernd Fischer found evidence for the Monster and Baby Monster simple groups, usually denoted M and B respectively; the Monster was discovered independently by Robert L. Griess. If g ∈ M then C_{M}(g), the centraliser in M of g, often contains a unique non-abelian composition factor, and in many cases this composition factor is a sporadic simple group. Indeed, using the notation for conjugacy classes given in the Atlas [4], when g ∈ 2A, 2B, 3A, 3B, 5B, 7A, 7B, 11A or 13A this composition factor is the Baby Monster B, the Conway group Co_{1}, the Fischer group Fi′_{24}, the sporadic Suzuki group Suz, the Hall–Janko group J_{2}, the Held group He, the alternating group A_{7}, the Mathieu group M_{12} or the linear group L_{3}(3) respectively. These groups were all known at the time; however, both C_{M}(3C) ≅ 3 × Th and C_{M}(5A) ≅ 5 × HN had non-abelian (sporadic) composition factors, now known as the Thompson group and the Harada–Norton group respectively, whose existence was previously unknown. Note that, following the Atlas, we write C_{M}(3C) to mean the centraliser in M of an element in class 3C, and so on. Koichiro Harada deduced much information about HN from knowledge of the involution centralisers in this putative group, namely 2·HS:2 and 2{1+8 \atop +}.(A_{5}×A_{5}).2, and Simon Norton [9] constructed HN as a permutation group on 1140000 points ‘by hand’, besides conducting a thorough investigation of its structure.

∗This project was supported by an EPSRC grant ref. GR/N27491/01.

Curtis [7] showed how consideration of a 15-dimensional 7-modular monomial representation of 3·A_{7}, the triple cover of the alternating group A_{7}, leads, in a well-motivated manner, to a definition of the Held group. As will be explained in more detail below, he obtained He as a homomorphic image of a split extension of a free product of 15 cyclic groups of order 7 by the group 3·A_{7}, an infinite semi-direct product which he denoted by 7^{⋆15}:_{m} 3·A_{7}. It was found to be more natural to extend this so-called progenitor to a group of shape 7^{⋆(15+15)}:_{m} 3·S_{7} in which both actions of A_{7} on 15 letters are realised, and the two sets of 15 elements of order 7 are interchanged by outer elements of S_{7}. A relation which forces certain pairs of these 30 symmetric generators of order 7 to generate copies of the simple group L_{2}(7) is sufficient to define He. It turns out that the much larger Harada–Norton group HN can be defined in a remarkably similar manner, this time working with generators of order 5. Specifically, we consider a 176-dimensional 5-modular representation of 2·HS, the double cover of the Higman–Sims group, to construct a progenitor of shape 5^{⋆176}:_{m} 2·HS. As in the Held case, it is more natural to ‘double up’ to a split extension of shape 5^{⋆(176+176)}:_{m} 2·HS:2 in which both actions of HS on 176 letters are realised, and outer elements interchange the two sets of 176 symmetric generators of order 5. As above, it turns out that a single relation which forces certain pairs of symmetric generators to generate copies of the simple group L_{2}(5) is sufficient to define HN. In both cases, He and HN, the action of the outer automorphism is easily described.

This approach is described below, and an outline of how it can be used to construct the 133-dimensional representation of HN over ℚ(\sqrt{5}) is given. When referring to this representation, the Atlas states that “explicit matrices have been computed”; however, such an explicit construction does not seem to occur in the literature. In his thesis, Norton [9] constructed 1140000 vectors which lay in a 133-dimensional subspace of a (1 + 462)-dimensional vector space over ℚ(\sqrt{5}). He specified the action of both A_{12}, which was reasonably straightforward, and an element g ∉ A_{12}, which was much harder, on these 1140000 vectors. In principle, we can convert the information in Norton’s thesis into 133 × 133 matrices generating HN, but this would involve considerably more work than what follows. Moreover, the construction employed here is well-motivated and could have resulted in the discovery of the group independently of the Monster. In effect, Norton constructed HN in the form HN = ⟨A_{12}, g⟩, using an 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}). However, we construct the group in the form HN = ⟨2·HS:2, t⟩ as an amalgam of 2·HS:2 and (D_{10} × U_{3}(5))·2 over U_{3}(5):4 ≅ (U_{3}(5) × 2)·2. In our approach, it turns out that maximal subgroups isomorphic to A_{12} emerge easily as subgroups generated by certain subsets of our symmetric generators.

The definition of HN as a homomorphic image of a progenitor of shape 5^{⋆(176+176)}:_{m} 2·HS:2 will be helpful, both in the construction of HN, and for verifying that we do have a representation of it. By-products are the permutation representations of HN of degrees 1140000 and 1539000 obtained by coset enumeration over A_{12} and 2·HS:2 respectively.

In Section 2, we introduce symmetric presentations, giving symmetric presentations of PGL_{2}(p) and L_{2}(p) for p a prime. We then describe how factoring ‘larger’ progenitors a relation that forces a certain ‘small’ subprogenitor to be PGL_{2}(p) or L_{2}(p) gives rise to the sporadic Held and Harada–Norton groups (just stated at this stage). This is intended to cover the analogy between the symmetric presentations of the Held group, already covered by Curtis [7], and the Harada–Norton group, which we discuss in this paper. It is also intended to convey the fact that it is a very ‘local’ relation that produces these symmetric presentations. In Section 3, we give presentations of HS:2 and 2·HS:2 that we shall need later. We also justify the fact that certain relations in these presentations are redundant.

In Section 4, we write down a presentation of the Harada–Norton group based on our [at this stage still conjectured] symmetric presentation. Coset enumeration proves the correctness of this presentation, and thus also of our symmetric presentation. This presentation also resolves the ambiguity that was present when we initially introduced the progenitor 5^{⋆(176+176)}:_{m} 2·HS:2 in Section 2.3 by providing a presentation for the progenitor. We end this section by deriving and describing all possible outer automorphisms of HN using our symmetric presentation along with some information about the conjugacy classes of HN.

In Section 5, we give alternative presentations of HN based on our symmetric presentation, and also related presentations of S_{7}, U_{3}(5):2 and HS:2. In Section 6, we determine all subgroups of HN generated subsets of the symmetric generators. We finish with Section 7 in which we briefly describe how to use our symmetric presentation to construct a 133-dimensional representation of HN over ℚ(\sqrt{5}).

2 Symmetric generation and progenitor groups

Following Curtis [6, 7], we adopt the notation m^{⋆n} to mean C_{m} ⋆ C_{m} ⋆ ⋯ ⋆ C_{m} (n times), a free product of n copies of the cyclic group of order m. Let F = T_{0} ⋆ T_{1} ⋆ ⋯ ⋆ T_{n−1} be such a group, with T_{i} = ⟨t_{i}⟩ ≅ C_{m}. Certainly permutations of the set of symmetric generators 𝒯 = {t_{0}, t_{1}, … t_{n−1}} induce automorphisms of F. But raising a given t_{i} to a power of itself coprime to m, while fixing the others, also gives rise to an automorphism of F. Together these generate the group M of all monomial automorphisms of F which is a wreath product H_{r} ≀ S_{n}, where H_{r} is an abelian group of order r = ϕ(m), the number of positive integers less than m and coprime to it. A semi-direct product of the form:

P ≅ m^{⋆n}:N,

where N is a subgroup of M which acts transitively on the set of cyclic subgroups 𝒯̄ = {T_{0}, T_{1}, …, T_{n−1}}, is called a progenitor. We call N the control subgroup and its elements monomial permutations or, more informally, monomials. Of course, N may simply permute the set of elements 𝒯 , as will always be the case when m = 2, and a wealth of interesting homomorphic images arise from this case. The more general case involving proper monomial action allows further fascinating possibilities. Note that, since P = ⟨N, 𝒯 ⟩ and the action of N on 𝒯 by conjugation is well-defined, elements of P may be put into a canonical form by gathering the elements of N on the left-hand side. Thus every element of P may be written, essentially uniquely, as an element of N followed by a word in the symmetric generators 𝒯 . In particular, if we seek homomorphic images of P, as we shall be doing, the relators by which we must factor will have form πw, for π ∈ N and w a word in the elements of 𝒯 .

As a classical example, we let p be a prime and consider

F = ⟨t_{1}⟩ ⋆ ⟨t_{2}⟩ ≅ C_{p} ⋆ C_{p} ≅ p^{⋆2}.

Then if λ is a generator of the cyclic group ℤ_{p}^{×}, the group of monomial automorphisms of F is isomorphic to C_{p−1} ≀ 2 and is generated by

\begin{matrix}π : t_{1} ↦ t_{1}^{λ}, t_{2} ↦ t_{2}, and \\ σ : t_{1} ↔ t_{2}. \end{matrix}

We abbreviate these monomial actions as

π ∼ ( \begin{matrix}λ & · \\ · & 1 \end{matrix} ) and σ ∼ ( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) ,

with the obvious meaning. Now it turns out that the projective general linear group PGL_{2}(p) is an image of

P ≅ p^{⋆2}:_{m} D_{2(p−1)},

where

N = ⟨ππ^{−σ}, σ⟩ = ⟨( \begin{matrix}λ & · \\ · & λ^{−1} \end{matrix} ) , ( \begin{matrix}· & 1 \\ 1 & · \end{matrix} )⟩ ≅ D_{2(p−1)}.

Indeed, the classical presentations of Todd [11], see also Stanley [10, page 130], in our language take the form:

\frac{p^{⋆2}:_{m} D_{2(p−1)}}{[( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) t_{1}]^{3}} ≅ PGL_{2}(p),  (1)

where the left-hand side denotes the progenitor p^{⋆2}:_{m} D_{2(p−1)} factored by the relator

[( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) t_{1}]^{3} ,

which may be rewritten as the relation

( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) = t_{1}t_{2}t_{1}.

We realise the image PGL_{2}(p) regarded as the group of linear fractional transformations of PG_{1}(p) = ℤ_{p} ∪ {∞} as follows:

ππ^{−σ} = ( \begin{matrix}λ & · \\ · & λ^{−1} \end{matrix} ) ∼ η ↦ λη and σ = ( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) ∼ η ↦ −\frac{1}{η}

and

t_{1} ∼ η ↦ η + 1 and t_{2} ∼ η ↦ \frac{η}{1 − η} .

For p odd, in order to obtain the simple group, we let

N = ⟨(ππ^{−σ})^{2}, σ⟩ = ⟨( \begin{matrix}μ & · \\ · & μ^{−1} \end{matrix} ) , ( \begin{matrix}· & 1 \\ 1 & · \end{matrix} )⟩ ≅ D_{p−1},

where μ (which we take, without loss of generality, to be λ^{2}) is a generator for the quadratic residues of ℤ_{p}^{×}. We then get:

\frac{p^{⋆2}:_{m} D_{p−1}}{[( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) t_{1}]^{3}} ≅ \{ \begin{matrix}L_{2}(p) × 2 & if p ≡ 1 (mod 4), \\ L_{2}(p) & if p ≡ 3 (mod 4). \end{matrix}  (2)

To quotient out the central involution in the case when p ≡ 1 (mod 4), we add the additional relator

[( \begin{matrix}· & λ^{−2} \\ λ^{2} & · \end{matrix} ) t_{1}^{λ}]^{3} .

Note that in all of the above presentations, whether the group we have presented is PGL_{2}(p), L_{2}(p) × 2 or L_{2}(p), we have ⟨t_{1}, t_{2}⟩ ≅ L_{2}(p).

Thus, if the two symmetric generators of order p are denoted by t_{1} and t_{2}, the control subgroup is generated by automorphisms of p^{⋆2} = ⟨t_{1}⟩⋆⟨t_{2}⟩. In Presentation 1 the element ππ^{−σ} conjugates t_{1} to t_{1}^{λ} and t_{2} to t_{2}^{λ^{−1}} and so acts as

(t_{1}, t_{1}^{λ}, t_{1}^{λ^{2}}, …, t_{1}^{λ^{−2}}, t_{1}^{λ^{−1}})(t_{2}, t_{2}^{λ^{−1}}, t_{2}^{λ^{−2}}, …, t_{2}^{λ^{2}}, t_{2}^{λ})

on the non-trivial powers of t_{1} and t_{2}; the involution σ interchanges t_{1} and t_{2} by conjugation, so we write that it has action (t_{1}, t_{2}). Note that we can determine the action of the given automorphisms on t_{i}^{j} for all i and j; in particular, (t_{1}, t_{2}) is an abbreviation for (t_{1}, t_{2})(t_{1}^{2}, t_{2}^{2}) … (t_{1}^{p−1}, t_{2}^{p−1}). The subscript ‘m’ on the colons in the progenitors above conveys the fact that the action is properly monomial.

Note that given an n × n monomial matrix A over ℤ_{m} where the nonzero entries of A are units, we can define the (group) action of A on F ≅ m^{⋆n} by t_{i}^{A} = t_{j}^{a_{ij}} where a_{ij} is the unique nonzero entry in the ith row of A.

2.1 Some notes and notation

We define N^{i} to be C_{N}(t_{i}) and N_{i} to be N_{N}(⟨t_{i}⟩). Note that N_{i} is the stabiliser of T_{i} in N, when we consider N acting as a permutation group on the T_{i}. Extending this notion, we define:

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

and

N_{i_{1}i_{2}...i_{r}} = N_{i_{1}} ∩ N_{i_{2}} ∩ ⋯ ∩ N_{i_{r}}.

Define a monomial matrix to be an m×m matrix over a ring R such that there is just one nonzero entry in each row and column, and a monomial group to be a group of monomial matrices. Note that in a monomial group the nonzero entries must be units of the ring R. Let {e_{i}} be a ‘basis’ for R^{m}. If our monomial group, G say, acts transitively on the ⟨e_{i}⟩ then this monomial representation of G is equivalent (up to conjugation by an invertible monomial matrix) to a representation induced up from a linear (i.e. 1-dimensional) representation of H = Stab_{G}(⟨e_{1}⟩) which has index m in G. Note that the different choices of transversal of H in G correspond to conjugations by monomial matrices (and in general, different transversals for induced representations correspond to conjugation by a ‘block-monomial’ matrix). If our monomial group G does not act transitively on the ⟨e_{i}⟩ then this monomial representation is the direct sum of representations induced from linear representations of two or more subgroups.

Often, we consider monomial representations where the nonzero entries are various roots of unity in ℂ. We then reduce these modulo p for various primes p in order to obtain representations over finite fields (which in the main cases of interest in this paper are always the prime fields ℤ_{p}). In general, the nonzero entries will correspond to elements of Aut(T), which we conveniently embed in the group ring ℤAut(T). The only such case when T is non-cyclic that we have considered extensively is T ≅ 2^{2} when we are sometimes led to consider monomial matrices over ℤS_{3}.

2.2 The Held progenitor and relation

In Curtis [7] we took N ≅ 3·S_{7} as our control subgroup. This group possesses subgroups of index 30 of shape 3 × L_{2}(7), and if we induce a non-trivial linear representation of such a subgroup up to N we obtain a (15 + 15)-dimensional faithful monomial representation of 3·S_{7} over any field which contains non-trivial cube roots of unity. We choose the field ℤ_{7} of integers modulo 7, and use this representation to define the action of N on the free product 7^{⋆(15+15)}. It can be shown that the two progenitors of shape 7^{⋆(15+15)}:3·S_{7} obtained by choosing 2 or 4 as our primitive cube root of unity are non-isomorphic. The subgroup of N fixing one of the symmetric generators, which is isomorphic to L_{2}(7), acts with orbits (1 + 14) + (7 + 8) on the 30 cyclic subgroups of order 7. Fixing a further symmetric generator in the 7-orbit is a subgroup isomorphic to S_{4} which acts with orbits (1 + 6 + 8) + (1 + 6 + 8). Normalising the subgroup generated by these two symmetric generators of order 7, r_{0} and s_{0} say, we have the ‘central’ element of order 3, which may be taken to square the r_{i} and fourth power the s_{i}, and an involution, commuting with the aforementioned S_{4}, interchanging them (replacing s_{0} by a power if necessary). The subgroup of N isomorphic to S_{4} mentioned above centralises r_{0} and s_{0}. Thus, in the notation used in Presentation 2, these two automorphisms of 7^{⋆2} are denoted by

( \begin{matrix}2 & · \\ · & 4 \end{matrix} ) and ( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) .

Therefore factoring by the relator

[( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) r_{0}]^{3}

ensures that ⟨r_{0}, s_{0}⟩ ≅ L_{2}(7), or an image thereof. Factoring one of the two progenitors of shape 7^{⋆(15+15)}:_{m} 3·S_{7} by a relator corresponding to this results in the Held group He, a sporadic simple group of order 4030 387200; with the other progenitor we get the trivial group. The outer automorphism of He is obtained by adjoining an element of order 2 which commutes with the control subgroup and inverts all the symmetric generators.

2.3 The Harada–Norton progenitor and relation

In order to define the Harada–Norton group HN in an analogous manner, using a progenitor with symmetric generators of order 5, we take as our control subgroup the group N of shape 2·HS:2 in which the outer involutions lift to elements of order 2. [The isoclinic variant of this group, namely 2·HS·2, has no outer involutions.] This group contains a subgroup H ≅ (2×U_{3}(5))·2 ≅ U_{3}(5):4, which is generated by U_{3}(5) together with an element of order 4 acting on it as on outer automorphism and squaring to the central involution. Thus, H/H′ ≅ C_{4}. In the usual way, we map a generator of H/H′ onto a primitive fourth root of unity in an appropriate field. We induce the corresponding linear representation of H up to N to obtain a faithful monomial (176 + 176)-dimensional representation of N. Over the complex numbers ℂ, this gives an irreducible representation whose restriction to 2·HS has character which is the sum of the two 176-dimensional characters given in Table 2 at the end of the paper. Of course, the field with fourth roots of unity which interests us is ℤ_{5}, which enables us to define a progenitor of shape 5^{⋆(176+176)}:_{m} 2·HS:2. As in the Held case, there are two non-isomorphic progenitors of this shape, depending on whether we choose 2 or 3 as our primitive fourth root of unity.

Let 𝒯 = ℛ ∪ 𝒮 = {r_{0}, r_{1}, …, r_{175}, s_{0}, s_{1}, …, s_{175}} be our set of symmetric generators, and let 𝒯̄ = {⟨r_{i}⟩, ⟨s_{i}⟩ : i = 0, 1, …, 175} be the set of cyclic subgroups they generate, arranged so that N′ ≅ 2·HS has orbits ℛ̄ = {⟨r_{0}⟩, ⟨r_{1}⟩, …, ⟨r_{175}⟩} and 𝒮̄ = {⟨s_{0}⟩, ⟨s_{1}⟩, …, ⟨s_{175}⟩} on 𝒯̄. Then H may be chosen to normalise ⟨r_{0}⟩, whence H′ commutes with ⟨r_{0}⟩, and both H and H′ have orbits (1 + 175) + (50 + 126) on 𝒯̄ = ℛ̄ ∪ 𝒮̄. We now choose ⟨s_{0}⟩ to lie in the 50-orbit of H (which is in 𝒮̄). Then ⟨r_{0}, s_{0}⟩ is centralised (in N) by a subgroup K of H′ isomorphic to A_{7}, which has orbits (1 + 7 + 42 + 126) + (1 + 7 + 42 + 126) on 𝒯̄. The subgroup ⟨r_{0}, s_{0}⟩ is normalised in N by an element of order 4 which squares the r_{i} and cubes the s_{i}, and an involution interchanging r_{0} and s_{0}. Together, these two elements extend K to N_{N}(K) ≅ (2^{2} ×A_{7}):2. Thus, the subgroup ⟨r_{0}, s_{0}⟩ ≅ 5^{⋆2} ≅ ⟨r_{0}⟩⋆⟨s_{0}⟩ is normalised in N by a subgroup isomorphic to D_{8}, which in the notation used in Presentation 1 acts as

( \begin{matrix}2 & · \\ · & 3 \end{matrix} ) and ( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) .

Therefore factoring by the relator

[( \begin{matrix}· & 1 \\ 1 & · \end{matrix} ) r_{0}]^{3} ,

where the matrix is one of the non-central involutions of N_{N}(K) ≅ (2^{2} × A_{7}):2 that commutes with K ≅ A_{7}, ensures that ⟨r_{0}, s_{0}⟩ ≅ L_{2}(5), or an image thereof. Factoring one of the two progenitors of shape 5^{⋆(176+176)}:_{m} 2·HS:2 by a relator corresponding to this results in the Harada–Norton group HN, a sporadic simple group of order 273 030912 000000. As before, factoring the other progenitor 5^{⋆(176+176)}:_{m} 2·HS:2 by the corresponding relation gives rise to the trivial group. The outer automorphism of HN is obtained by adjoining an element of order 4 which commutes with N′ and squares all the symmetric generators (and is thus inverted by the elements in N \ N′).

2.4 Non-isomorphic progenitors

In Bray [1, 2], it is shown that the two (possible) progenitors 5^{⋆(176+176)}:_{m} 2·HS:2 in which the central involution of N inverts all the symmetric generators are non-isomorphic. This involved a double coset count, and also shows that for each p ⩾ 13, with p prime and p ≡ 1 (mod 4) there are two non-isomorphic progenitors p^{⋆(176+176)}:_{m} 2·HS:2 in which N ≅ 2·HS:2 acts faithfully. The same technique also shows that for each prime p ⩾ 7 with p ≡ 1 (mod 6) there are two isomorphism types of progenitor p^{⋆(15+15)}:_{m} 3·S_{7} in which the control subgroup acts faithfully.

Naturally, the two similar-looking progenitors are rather difficult to distinguish. One technique to do this might be to introduce an invariant. The ‘invariant’ we want to take is the ‘abelianisation’ of the progenitor. The version of ‘abelianisation’ we want is rather weaker than the usual one, namely we just make T_{i} and T_{j} commute whenever i ≠ j; thus the ‘abelianisation’ of the progenitor T^{⋆n}:N in this sense is T^{n}:N. Unfortunately, this definition can depend on the generating set chosen and so is not an invariant of the group. This problem is also addressed in Bray [2].

In our case, we can rescue the situation, for if P ≅ p^{⋆(176+176)}:_{m} 2·HS:2 and ϕ : P → N is an epimorphism then ker ϕ is forced to be the p^{⋆352} we first thought of. To see this, let ζ be the central involution of N and firstly suppose that ζϕ = 1. Since ζ inverts all the symmetric generators this forces tϕ = 1 for all symmetric generators t and thus forces Pϕ to be an image of HS:2, a contradiction. Thus ζϕ ≠ 1 and Nϕ ≅ N, which forces Pϕ = Nϕ = N. But now N^{0}ϕ ≅ U_{3}(5) has no elements of order p in its N-centraliser; thus t_{0}ϕ = 1 since t_{0}ϕ commutes with N^{0}ϕ and t_{0}ϕ has order 1 or p. Therefore tϕ = 1 for all symmetric generators t. Similar arguments show that the only epimorphisms from p^{⋆(15+15)}:_{m} 3·S_{7} onto 3·S_{7} have the visible p^{⋆(15+15)} in their kernels. In these cases, our abelianisation consists of performing the usual group theoretic abelianisation on the ker ϕ’s, which we have now established to be unique.

If p ⩾ 13, with p prime and p ≡ 1 (mod 4), then both progenitors p^{⋆(176+176)}:_{m} 2·HS:2 have isomorphic abelianisations p^{352}:2·HS:2, and if p ⩾ 13, with p prime and p ≡ 1 (mod 6), then both progenitors p^{⋆(15+15)}:_{m} 3·S_{7} have isomorphic abelianisations p^{30}:3·S_{7}. In each case, N ≅ 2·HS:2 acts irreducibly on the p^{352} regarded as an ℤ_{p}N-module; also 3·S_{7} acts irreducibly on the p^{30} regarded as an ℤ_{p}3·S_{7}-module. The 5^{⋆(176+176)}:_{m} 2·HS:2 giving rise to the Harada–Norton group abelianises as 5^{240·56·56}:2·HS:2, with submodules of dimensions 0, 240, 296 and 352 only; the other 5^{⋆(176+176)}:_{m} 2·HS:2 abelianises as 5^{56·56·240}:2·HS:2 [with the dual module]. The abelianisation of the progenitor 7^{⋆(15+15)}:_{m} 3·S_{7} giving rise to the Held group is 7^{12·18}:3·S_{7} with a single non-trivial proper submodule of dimension 12. The other 7^{⋆(15+15)}:_{m} 3·S_{7} has abelianisation 7^{18·12}:3·S_{7}.

3 The Higman–Sims group

We must first find a presentation for our proposed control subgroup 2·HS:2. Our starting point is the symmetric presentation of Curtis [6] given below:

\frac{2^{⋆50}:(U_{3}(5):2)}{t_{i}t_{k}t_{i}t_{j} = (i, k)} ≅ HS:2,  (3)

where i—j—k is a path in the Hoffman–Singleton graph and (i, k) is the unique non-trivial element of U_{3}(5):2 which commutes with the stabiliser of i and k. This is an involution, and is also the unique element of U_{3}(5):2 that fixes j, swaps i and k, and fixes the other five points joined to j. Thus (i, k) is a transposition in the copy of S_{7} fixing j.

So that we can successfully complete the required coset enumeration for HN, we have opted to create an alternative presentation of HS:2 based on the above symmetric presentation, rather than the one given in [6]. As generators of H ≅ U_{3}(5):2, we choose x of order 2 and y of order 6 such that x is an outer involution playing the rôle of (i, k) in Presentation 3, and ⟨x, y^{2}⟩ ≅ S_{7}. These can be chosen to satisfy the presentation:

⟨x, y | x^{2} = y^{6} = [x, y]^{3} = (xy^{2}xy^{3})^{3} = (xy)^{8} = [x, y^{−2}xyxyxy^{−2}] = (xyxy^{3})^{4} = 1⟩, (4)

which can readily be shown to define U_{3}(5):2. If the relation [x, y^{−2}xyxyxy^{−2}] = 1 is omitted, the resulting presentation defines 3·U_{3}(5):2, in which the outer elements invert the ‘central’ elements of order 3. In order to try to keep the length of the additional relation t_{i}t_{k}t_{i}t_{j} = (i, k) to a minimum, we choose z to correspond to t_{j}, whence its centraliser in H is ⟨x, yxyxy^{2}xy^{−1}⟩ ≅ S_{7}. The symmetric generator t_{i} is then given by z^{y^{3}xyxy}, and the additional relation of Presentation 3 reduces to (z^{y^{3}xyxy}x)^{3} = z. This leads to the following presentation for HS:2:

\begin{matrix}N = ⟨x, y, z | x^{2} = & y^{6} = [x, y]^{3} = (xy^{2}xy^{3})^{3} = (xy)^{8} = [x, y^{−2}xyxyxy^{−2}] = (xyxy^{3})^{4} \\ = & z^{2} = (xz)^{2} = [z, yxyxy^{2}xy^{−1}] = z(z^{y^{3}xyxy}x)^{3} = 1⟩ ≅ HS:2, \end{matrix} (5)

which can be verified by performing a coset enumeration over H = ⟨x, y⟩, to obtain the index 352. In fact, the relation [x, y^{−2}xyxyxy^{−2}] = 1 of Presentation 5 is redundant. This comes about because coset enumeration over ⟨x, y⟩ still yields the index 352, and so the derived subgroup of N, which is still perfect, has shape 3.HS or HS. In the former case, the central 3 of N′ would be generated by the image of [x, y^{−2}xyxyxy^{−2}], which is inverted by x. But this is a contradiction since ⟨x, y⟩ ⩽ N′. Thus, the latter case (i.e. N′ ≅ HS) must hold, and since it is easily seen, by abelianising the presentation, that N′ has index 2 in N, we must have N ≅ HS:2. Thus, we have shown:

\begin{matrix}N = ⟨x, y, z | x^{2} = & y^{6} = [x, y]^{3} = (xy^{2}xy^{3})^{3} = (xy)^{8} = (xyxy^{3})^{4} \\ = & z^{2} = (xz)^{2} = [z, yxyxy^{2}xy^{−1}] = z(z^{y^{3}xyxy}x)^{3} = 1⟩ ≅ HS:2. \end{matrix}  (6)

3.1 The double cover of HS:2

We must now construct a presentation of the double cover 2·HS:2. We shall let x, y, z, H and N refer to preimages in 2·HS:2 (Atlas version) of what they were in HS:2. 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. Thus, to derive a presentation of 2·HS:2 from our presentation of HS:2, we must first ensure that x^{2} is central of order 2, and then set each of the relators in Presentation 5 equal to 1 or x^{2} as appropriate. Without loss of generality, we may assume that y ∈ H′ (since this happens in the image HS:2). Since H/H′ = ⟨xH′⟩, with yH′ = H′, we can determine whether a relator which is the identity in the image U_{3}(5):2 is still the identity in H, or whether it is the central involution of H. Thus, we easily determine that

\begin{matrix}⟨x, y | x^{4} = & [x^{2}, y] = y^{6} = [x, y]^{3} = (xy^{2}xy^{3})^{3}x^{−2} \\ = & (xy)^{8} = [x, y^{−2}xyxyxy^{−2}] = (xyxy^{3})^{4} = 1⟩, \end{matrix}

is a presentation for H ≅ U_{3}(5):4. Now z is the preimage of an element of class 2C and so has order 2. The centraliser of the image of z in the image of H is a subgroup isomorphic to S_{7}. It turns out that the preimage of this copy of S_{7}, which has shape (A_{7} × 2)·2, has elements that conjugate z to zx^{2}. Thus z commutes with the unique subgroup of index 2 in this preimage, which has shape A_{7} × 2. This implies that z inverts x, i.e. (zx)^{2} = 1 holds, and also that yxyxy^{2}xy^{−1} conjugates z to zx^{2} (since this element is in the outer half of the copy of (A_{7} × 2)·2 referred to above), and so we have z^{2} = (zx)^{2} = [z, yxyxy^{2}xy^{−1}]x^{−2} = 1. These relations imply that [z, x^{2}] = 1 and [z, x] = x^{2}. The map (x, y, z) ↦ (x^{−1}, y, z) fixes all the previous relations but interchanges the two preimages of the relation z(z^{y^{3}xyxy}x)^{3} = 1 of HS:2, and so z(z^{y^{3}xyxy}x)^{3} can be set equal to either of the central elements; naturally we choose z(z^{y^{3}xyxy}x)^{3} = 1. We may check that we have not factored out the central element of 2·HS:2 by taking a permutation representation of degree 1408 over ⟨y, y^{x}⟩ ≅ U_{3}(5). Thus we have:

\begin{matrix}⟨x, y, z | x^{4} = & [x^{2}, y] = y^{6} = [x, y]^{3} = (xy^{2}xy^{3})^{3}x^{−2} = (xy)^{8} = [x, y^{−2}xyxyxy^{−2}] \\ = & (xyxy^{3})^{4} = z^{2} = (zx)^{2} = [z, yxyxy^{2}xy^{−1}]x^{−2} = z(z^{y^{3}xyxy}x)^{3} = 1⟩ ≅ 2·HS:2. \end{matrix}

In order to shorten the above presentation for 2·HS:2, we first replace x^{4} = [x^{2}, y] = 1 = (xy^{2}xy^{3})^{3}x^{−2} by x^{2}yx^{2}y^{−1} = 1 = (x^{−1}y^{2}xy^{3})^{3} in the preceding presentation of (U_{3}(5) × 2)·2. Furthermore, in either of these presentations of (U_{3}(5) × 2)·2, omitting [x, y^{−2}xyxyxy^{−2}] = 1 gives rise to (3·U_{3}(5) × 2)·2. In particular,

⟨x, y, z | x^{2}yx^{2}y^{−1} = y^{6} = [x, y]^{3} = (x^{−1}y^{2}xy^{3})^{3} = (xy)^{8} = (xyxy^{3})^{4} = 1⟩  (7)

is a presentation for (3·U_{3}(5) × 2)·2, as can verified by coset enumeration over ⟨xy⟩ ≅ C_{8}. We then verify equality between ⟨x, yxyxy^{2}xy^{−1}⟩ and ⟨x, yxy^{−1}xy, (yxy)^{2}, y^{2}xy^{−2}xy^{2}⟩ by using coset enumeration with Presentation 7. An argument similar to the one we employed in the HS:2 case shows that:

\begin{matrix}⟨x, y, z | x^{2}yx^{2}y^{−1} = & y^{6} = [x, y]^{3} = (x^{−1}y^{2}xy^{3})^{3} = (xy)^{8} = (xyxy^{3})^{4} = z^{2} \\ = & (zx)^{2} = [z, yxyxy^{2}xy^{−1}]x^{−2} = z(z^{y^{3}xyxy}x)^{3} = 1⟩ ≅ 2·HS:2. \end{matrix} (8)

It will be convenient in what follows to use the notation N_{0} = H and N^{0} = H′ to denote the normaliser of ⟨r_{0}⟩ and the centraliser of r_{0} respectively.

4 The progenitor and extra relation

We are now in position to write down a presentation for a progenitor of shape

5^{⋆(176+176)}:_{m} 2·HS:2,

in which the central element of N ≅ 2·HS:2 inverts all the symmetric generators. This is done by adding a generator t (= r_{0}) of order 5 to Presentation 8, and requiring that t commutes with N^{0} = ⟨y, y^{x}⟩ and t^{x} = t^{2} or t^{3}. This last choice must result in nonisomorphic progenitors, since we know that there are two possible such progenitors up to isomorphism.

So far, we have the additional relations t^{5} = [t, y] = t^{x}t^{±2} = 1; we must now seek an extra relation that forces ⟨r_{0}, s_{0}⟩ ≅ L_{2}(5), where ⟨s_{0}⟩ lies in the 50-orbit of 𝒯̄ under N_{0}. We may take s_{0} = r_{0}^{z}. Referring to Lemma 1 of Curtis [5], we find that ⟨r_{0}, s_{0}⟩ ∩ N ⩽ C_{N}(C_{N}(r_{0}) ∩ C_{N}(s_{0})) = ⟨z, x^{2}⟩. Thus our two choices for the extra relation are (zt)^{3} = 1 or (zx^{2}t)^{3} = 1, since x^{2} inverts t. But (zt)^{3} = 1 implies that (zx^{2}t^{2})^{3} = 1, for there are elements of N that conjugate z to zx^{2} and t to t^{2}, and we have only specified t up to powering. So we may take (zt)^{3} = 1 as our final relation.

For both of the progenitors 5^{⋆(176+176)}:_{m} 2·HS:2 in which we were 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} we obtain |G : N| = 1; this is sufficient to show that G = 1 in this case. In the case when t^{x} = t^{−2} we find that |G : N| = 1539000, which is the index we seek. In performing these enumerations, we appended some redundant relations. The actual presentations we used are:

\begin{matrix}⟨x, y, z, t | x^{4} = & [x^{2}, y] = y^{6} = [x, y]^{3} = (xy^{2}xy^{3})^{3}x^{−2} = (xy)^{8} = [x, y^{−2}xyxyxy^{−2}] \\ = & (xyxy^{3})^{4} = z^{2} = (xz)^{2} = z(z^{y^{3}xyxy}x)^{3} = [z, yxy^{−1}xy] = [z, (yxy)^{2}] \\ = & [z, y^{2}xy^{−2}xy^{2}] = t^{5} = t^{x}t^{±2} = [t, y] = (zt)^{3} = 1⟩. \end{matrix}

Putting all of this together, we obtain that:

\begin{matrix}⟨x, y, z, t | y^{6} = & x^{2}yx^{2}y^{−1} = [x, y]^{3} = (x^{−1}y^{2}xy^{3})^{3} = (xy)^{8} = (xyxy^{3})^{4} = z^{2} = (zx)^{2} \\ = & [z, yxyxy^{2}xy^{−1}]x^{−2} = z(z^{y^{3}xyxy}x)^{3} = t^{5} = t^{x}t^{2} = [t, y] = (zt)^{3} = 1⟩. \end{matrix}

is a presentation for the Harada–Norton group HN.

Note. We invariably performed coset enumerations in Magma [3] with Hard:=true set. The coset enumerations we have been required to perform up to this point (the HNenumerations described above, the enumeration of the cosets of ⟨x, y⟩ in Presentation 8, and so on) have not required more than 2 million cosets to complete. Some other presentations, such as those for HN with the redundant relations removed, might require more cosets to be defined in their enumerations. The enumerations associated with Section 5 can also be performed using relatively small tables, with the enumerations of the HN groups over their visible 2·HS:2 subgroups requiring no more than 2 million cosets (and no redundant relations).

4.1 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 subgroups of shape 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}. Thus a now centralises N^{0} and so a normalises C_{HN}(N^{0}) = ⟨t, x^{2}⟩ ≅ D_{10}. Therefore 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}. Thus we easily get two maximal subgroups of HN:2 as follows: ⟨N, a⟩ ≅ 4·HS:2 and ⟨N_{0}, t, a⟩ ≅ 5:4 × U_{3}(5):2. The automorphism a squares the r_{i} and cubes the s_{i}.

We remark that the above derives the form of the outer automorphism of HN and we remark also that the element we give above is indeed an outer automorphism. This uses the class list of HN and information we provide via our symmetric presentation. Thus we have shown that Out(HN) ≅ 2. Note that similar arguments, applied to the Held symmetric presentation of Curtis [7], show that Out(He) ≅ 2 and moreover derive the outer automorphism exhibited in that paper.

5 Another presentation of HN

More recently, we have produced another presentation of HN which is also derived from our symmetric presentation and which may be of interest to the reader. We include presentations of HS:2, U_{3}(5):2 and S_{7} as well. Implicit within this series of presentations are HS:2 as an image of 2^{⋆50}:(U_{3}(5):2) in the aforementioned manner, and U_{3}(5):2 as an image of a progenitor of shape 2^{⋆42}:S_{7}. We remark that in each of the presentations of HN given below a subgroup isomorphic to A_{12} is given by ⟨α, β, δ, δ^{γβγ}, τ⟩; in the first presentation, this subgroup also contains ζ.

\begin{matrix}⟨α, β, γ, δ, ζ, τ | ζ^{2} = & [ζ, α] = [ζ, β] = [ζ, γ] = [ζ, δ] = α^{2}ζ = β^{7} = (αβ^{2})^{4} \\ = & (αβαβ^{3})^{3}ζ = ((αβ)^{3}αβ^{−3})^{2} = γ^{2}ζ = [α, γ] = [βαβ, γ] \\ = & (βαβ^{3}γ)^{3} = δ^{2} = (αδ)^{2} = [β, δ] = δ^{γβγβ^{−1}}(γδ)^{3} \\ = & τ^{5} = (ζτ)^{2} = τ^{α}τ^{2} = τ^{γ}τ^{−2} = [τ, β] = (δτ)^{3} = 1⟩ ≅ HN. \end{matrix}

\begin{matrix}⟨α, β, γ, δ, τ | α^{4} = & [α^{2}, β] = β^{7} = (αβ^{2})^{4} = α^{−2}(αβαβ^{3})^{3} = ((αβ)^{3}αβ^{−3})^{2} \\ = & γ^{2}α^{2} = [α, γ] = [βαβ, γ] = (βαβ^{3}γ)^{3} = δ^{2} = (αδ)^{2} = [β, δ] \\ = & δ^{γβγβ^{−1}}(γδ)^{3} = τ^{5} = τ^{α}τ^{2} = τ^{γ}τ^{−2} = [τ, β] = (δτ)^{3} = 1⟩ ≅ HN. \end{matrix}

\begin{matrix}⟨α, β, γ, δ | α^{2} = & β^{7} = (αβ^{2})^{4} = (αβαβ^{3})^{3} = ((αβ)^{3}αβ^{−3})^{2} = γ^{2} = [βαβ, γ] \\ = & [α, γ] = (βαβ^{3}γ)^{3} = δ^{2} = [α, δ] = [β, δ] = δ^{γβγβ^{−1}}(γδ)^{3} = 1⟩ ≅ HS:2. \end{matrix}

\begin{matrix}⟨α, β, γ | α^{2} = & β^{7} = (αβ^{2})^{4} = (αβαβ^{3})^{3} = ((αβ)^{3}αβ^{−3})^{2} \\ = & γ^{2} = [α, γ] = [βαβ, γ] = (βαβ^{3}γ)^{3} = 1⟩ ≅ U_{3}(5):2. \end{matrix}

⟨α, β | α^{2} = β^{7} = (αβ^{2})^{4} = (αβαβ^{3})^{3} = ((αβ)^{3}αβ^{−3})^{2} = 1⟩ ≅ S_{7}.

6 Subgroups of HN generated by subsets of the symmetric generators

In this section, we are interested in subgroups generated by subsets of the symmetric generators. Since for t a symmetric generator, the four elements t, t^{2}, t^{3} and t^{4} each generate the cyclic subgroup ⟨t⟩, the relevant objects to consider are ⟨t⟩ for t a symmetric generator. Up to now, we have not given a name to the subgroups ⟨t⟩; in this paper, we shall refer to them as cogs. The progenitor 5^{⋆(176+176)}:_{m} 2·HS:2 contains 352 cogs, and each cog contains 4 symmetric generators. We shall label a cog by a symmetric generator it contains.

6.1 Subgroups generated by up to 2 cogs

Firstly, we note that 0 and 1 cogs generate the cyclic groups of orders 1 and 5 respectively. Given a cog t, N_{N}(t) ≅ 2.(U_{3}(5):2) has orbits 1 + 175 + 50 + 126 on cogs, the orbit of size 1 being {t}.

If u is in the orbit of size 50 under N_{N}(t), we say that t and u are 50-joined and we have ⟨t, u⟩ ≅ A_{5}. This subgroup contains no further cogs. If u is in the orbit of size 126 under N_{N}(t), we say that t and u are 126-joined and we have ⟨t, u⟩ ≅ 5{1+2 \atop +}. This subgroup contains no further cogs.

If u is in the orbit of size 175 under N_{N}(t), we say that t and u are 175-joined and we have ⟨t, u⟩ ≅ A_{6}. In this case, unlike the two cases above, t and u are cogs from the same side, that is N′-orbit; thus t and u are both in ℛ̄ or both in 𝒮̄. The subgroup ⟨t, u⟩ contains two cogs from the other side, v and w say, which we call the mates of the first two. The two mates v and w are 175-joined; the four joins between {t, u} and {v, w} are all 50-joins. The group ⟨t, u⟩ contains just 4 cogs.

Note that if t and u are on the same side then the stabiliser in N of {t, u}, a group of shape 2.(2 × A_{6}.2^{2}), has two orbits of size 2, namely {t, u} and {v, w}, with {v, w} being the mates of {t, u} as above.

6.2 Subgroups generated by 3 or more cogs

In each of the cases below, the subgroup generated by the cogs under consideration contains the ‘matiness closure’ of those cogs, as it must, and no further cogs.

If a subgroup contains 3 or more cogs then two of them are on the same side, so that the subgroup contains a {t, u, v, w} configuration generating an A_{6} as detailed above. The stabiliser in N of such a configuration is a maximal subgroup of shape 2.(2 × A_{6}.2^{2}).2. This subgroup has orbits of lengths 4 + 24 + 144 + 180 on cogs, with the orbit of length 4 being {t, u, v, w}. All of these orbits split equally across ℛ̄ and 𝒮̄; thus the extra cog we add, s say, can be assumed to be on the same side as t and u.

If s is in the 180-orbit, so that s is 126-joined to both v and w, then ⟨t, u, v, w, s⟩ ≅ 2·HS. This subgroup contains 32 = 16 + 16 cogs, and these happen to be the matiness closure of the 5 given cogs. The stabiliser in N of such a configuration of 32 cogs is a maximal subgroup of shape 2.(2{1+6 \atop +}:S_{5}). If we add in yet another cog not in these 32 then we obtain the whole of HN, which of course contains all the cogs, and thus all of the symmetric generators. In fact, the stabiliser of these 32 cogs acts transitively on the remaining 320 cogs; adding one of these to our 32 and taking the matiness closure gives all the cogs.

If s is in the 144-orbit, so that s is 50-joined to one of v and w, and 126-joined to the other, then ⟨t, u, v, w, s⟩ ≅ U_{3}(5). This subgroup contains 12 = 6 + 6 cogs, and these are the matiness closure of the 5 given cogs. The stabiliser in N of this configuration of 12 cogs is a maximal subgroup of shape 2.(5:4×S_{5}), and this has orbits 12+100+240 on the cogs, with 12-orbit being the subgroup generating U_{3}(5). Taking these 12 cogs plus a cog from the 240-orbit yields a configuration generating the whole of HN; indeed the matiness closure of this configuration contains all cogs. Taking the 12 cogs plus a cog from the 100-orbit yields a configuration generating 2·HS; we have already seen this configuration above, and the matiness closure of the 12 + 1 cogs consists of all 32 cogs that lie in this particular copy of 2·HS. Note that both the 100-orbit and the 240-orbit intersect the aforementioned 180-orbit non-trivially, so that no new cases can arise by extending these 12 cogs.

In the final case, s is in the 24-orbit, so that s is 50-joined to both v and w. Thus any case in which 126-joins occur has already been dealt with in one of the two cases above, so that from now on we may assume that all of our joins are 50-joins or 175-joins. Then we have ⟨t, u, v, w, s⟩ ≅ A_{7}, and this contains just one more cog, s′ say. In fact, there are just 10 cogs that can ‘validly’ extend these 6; adding all 10 of them gives A_{12} (with no 126-joins present). Adding the cogs one at a time to A_{7} gives A_{8}, A_{9}, A_{10}, A_{11} and A_{12}. At each stage another cog lies in the subgroup generated by the previous configuration and the just-added cog; thus we append two cogs to the configuration at each stage. For n ∈ {7, 8, 9, 10, 11, 12}, A_{n} is generated (in a unique way) by a configuration of 2(n − 4) cogs whose [set-wise] stabiliser in N, namely 2.(2 × S_{12−n} × S_{n−4}), acts transitively upon them, and also acts transitively upon the 2(12 − n) cogs that can be validly added to the configuration. This completes the analysis of all subgroups generated by subsets of the symmetric generators.

6.3 Table of results

In Table 1 below we display information about all subsets of symmetric generators (up to N-conjugacy) whose image in HN contains no further symmetric generators. In the table below, H denotes the subgroup generated by the symmetric generators, and the “maximal” column refers to the maximality of ⟨H, N_{N}(H)⟩ in HN. An m_{1} + m_{2} in the “config[uration] code” column tells how many cogs H has on each side. An entry m + m (50) in the “config. code” column indicates that there are no 126-joins between the cogs of H; an entry m + m (126) indicates that there is at least one 126-join.

In the 6 + 6 (126) configuration, each cog is 126-joined to just one other [the ‘126-graph’ of this configuration is six copies of K_{2}]. In the 16 + 16 (126) configuration, each cog is 126-joined to just six others. The associated 126-graph has subsets of {0, 1, 2, 3, 4, 5} modulo complementation as vertices, and S is joined to S ∪ {i} if i ∉ S and to S \ {i} if i ∈ S.

Table 1: Subgroups of HN generated by subsets of the symmetric generators

config. code  H  N_{N}(H)  H ∩ N  ⟨H, N_{N}(H)⟩ maximal
0 + 0  1  2.(HS:2)  1  2·HS:2  yes
1 + 0  5  2.(U_{3}(5):2)  1  (D_{10} × U_{3}(5))·2  yes
1 + 1 (50)  A_{5}  2.(2 × S_{7})  2^{2}  (A_{5} × A_{7}):2  no
1 + 1 (126)  5{1+2 \atop +}  2.(5{1+2 \atop +}:[2^{5}])  5  5{1+4 \atop +}:[2^{6}]  no
2 + 2 (50)  A_{6} 2.(2 × A_{6}.2^{2}).2  D_{8}  (A_{6} × A_{6}).2^{2}  no
3 + 3 (50)  A_{7} 2.(2 × S_{5} × S_{3}) (2^{2} × 3):2  (A_{7} × A_{5}):2  no
4 + 4 (50)  A_{8} 2.(2 × S_{4} × S_{4}) (2^{2} × A_{4}):2 (A_{8} × A_{4}):2  no
5 + 5 (50)  A_{9} 2.(2 × S_{3} × S_{5}) (2^{2} × A_{5}):2  (A_{9} × 3):2  no
6 + 6 (50)  A_{10}  2.(2^{2} × S_{6}) (2^{2} × A_{6}):2  S_{10}  no
7 + 7 (50)  A_{11}  2.(2 × S_{7})  (2^{2} × A_{7}):2  A_{11}  no
8 + 8 (50)  A_{12}  2.(2 × S_{8})  (2^{2} × A_{8}):2  A_{12}  yes
6 + 6 (126)  U_{3}(5) 2.(5:4 × S_{5})  2·S_{5}^{+}  (D_{10} × U_{3}(5))·2  yes
16 + 16 (126) 2·HS  2.(2{1+6 \atop +}:S_{5})  2.4.2^{4}.S_{5}  2·HS:2  yes
176 + 176 (all) HN  2.(HS:2)  2·HS:2  HN  no

The symmetric presentation of U_{3}(5):2 related to the above 6 + 6 (126) configuration is given by Curtis [6] in his Higman–Sims paper. Jabbar [8, pp. 133–134] has produced a presentation for HS as a homomorphic image of 5^{⋆(16+16)}:_{m} 4.2^{4}:S_{5}; this is the symmetric presentation satisfied by the 16 + 16 (126) configuration above.

The 8 + 8 (50) configuration that generates A_{12} gives rise to a symmetric presentation of A_{12} as follows. Let t_{i} ∼ (i, 8, 9, X, ∞) and u_{i} ∼ (i, 9, 8, ∞, X) for 0 ⩽ i ⩽ 7. Then the t_{i} and u_{i} generate A_{12}. The symmetries in S_{12} of the t_{i} and u_{i} consist of: Sym({0, …, 7}), permuting the t_{i} and u_{i} naturally; σ ∼ (8, 9)(X, ∞), which swaps t_{i} and u_{i} (for all i); and ι ∼ (8, 9, ∞, X), which squares the t_{i} and cubes the u_{i}. These symmetries generate a subgroup isomorphic to S_{8} × D_{8}, of which (A_{8} × 2^{2}).2 ≅ \frac{1}{2}(S_{8} × D_{8}) lies in A_{12}. Thus A_{12} is an image of the progenitor 5^{⋆(8+8)}:_{m} (A_{8} × 2^{2}).2. Using mechanical coset enumeration we verify that:

\frac{5^{⋆(8+8)}:_{m} (A_{8} × 2^{2}).2}{(σt_{0})^{3} = ((0, 1)ισt_{0}^{2})^{3} = 1} ≅ A_{12}.  (9)

In fact, the extra relation (zt)^{3} we append to 5^{⋆(176+176)}:_{m} 2·HS:2 implies that both of the extra relations (σt_{0})^{3} = ((0, 1)ισt_{0}^{2})^{3} = 1 hold in a subprogenitor 5^{⋆(8+8)} with 8 + 8 (50) configuration; thus coset enumeration over this subgroup, which has index 1140000, is also a valid method to establish that we have a symmetric presentation of HN. In the notation of Section 4, a copy of A_{12} generated in the above manner is given by ⟨x, yxyxy^{2}xy^{−1}, z, z^{yxy}, t⟩.

The above symmetric presentation of A_{12} belongs to a series of symmetric presentations of the alternating groups by replacing 8 by n and 12 by n + 4 for n ⩾ 4, see Bray [1]. We shall investigate this series of symmetric presentations, along with related ones, elsewhere.

The essence of the calculations in this section is to find all matiness closed sets of cogs. These calculations were done (with the aid of a computer) using the 352-point permutation action of HS:2 (the central involution of N fixes all of the cogs). The cog-wise stabiliser of two 175-joined cogs has orbits 1 + 1 + 12 + 72 + 90 + 2 + 12 + 36 + 36 + 90 on cogs; the mates of the two given cogs form the orbit of size 2. The N_{N}(H) column of Table 1 was also compiled using this permutation representation. Calculations within subprogenitors, in particular to verify that ⟨x, yxyxy^{2}xy^{−1}, z, z^{yxy}, t⟩ (notation as in Section 4) satisfies Symmetric Presentation 9, can be done using the 1408 point action of N ≅ 2·HS:2. We then work in A_{12} to verify the assertion that the subgroup generated by two 175-joined cogs contains just two further cogs. We use our knowledge of A_{12} to fill in most of the rest of the table. We can use, say, a 133-dimensional representation of HN over 𝔽_{9} to help us determine what the configurations 1 + 1 (126), 6 + 6 (126) and 16 + 16 (126) generate. (The 133-dimensional representation we make below is written over the ring ℤ[\sqrt{5}, \frac{1}{2}, \frac{1}{5}], and so can be directly reduced modulo 3.)

7 The matrix construction

We now give an outline of how to use our symmetric presentation to construct the faithful 133-dimensional representation(s) of HN over ℚ(\sqrt{5}); these are interchanged by the field automorphism \sqrt{5} ↦ −\sqrt{5}. Such a representation restricts to 2·HS:2 as 77a^{+} ⊕ 56a^{±}. The two possible 56-dimensional representations are faithful and differ by only the sign of \sqrt{5} and so we may choose either one.

The 77-dimensional representation of [2·]HS:2 was chopped out of a 100-dimensional permutation module (along with a 22-dimensional representation), so this was relatively easy to obtain. Constructing the 56-dimensional representation of 2·HS:2 was much harder. We shall omit the details of how this was done, but note that it was aided by the symmetric presentation of HS:2 as an image of 2^{⋆50}:(U_{3}(5):2) that we have already mentioned in this paper. The details are contained in the first author’s PhD thesis, see Bray [1]. Care should be taken when reading Bray [1] since the generators that were chosen for N_{0} then are different from those used in this paper. A website available at:

http://web.mat.bham.ac.uk/spres/

contains, among other things, a 56-dimensional representation of 2·HS:2 over ℤ[\sqrt{5}, \frac{1}{5}] and a 133-dimensional representation of HN over ℤ[\sqrt{5}, \frac{1}{2}, \frac{1}{5}]. These representations are given on the generators x, y, z and t used in this paper.

The restriction to N^{0} ≅ U_{3}(5) is as (21a ⊕ 28b ⊕ 28c) ⊕ (28b ⊕ 28c). (The labelling of the 28-dimensional irreducibles of N^{0} is chosen so that the permutation module of N^{0} over C_{N^{0}}(z) ≅ A_{7} is 1a ⊕ 21a ⊕ 28a.) With respect to a well-chosen 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. It is to be understood that, for each element of N^{0}, the submatrices in each of the 28b blocks are the same; the same applies to the 28c blocks. Again, the first author’s PhD thesis tells us how to produce such a desirable basis.

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 \sqrt{5}. 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\} = \{\frac{1}{2}(−1 + \sqrt{5}), \frac{1}{2}(−1 − \sqrt{5})\} and without loss of generality we may set β = 1, since we may conjugate t by elements centralising N. However, our matrices for N are in a reasonably ‘nice’ form and the map induced by applying the field automorphism \sqrt{5} ↦ −\sqrt{5} 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.

We remark (without proof) that the minimal degree of a true representation (see next sentence) of our progenitor 5^{⋆(176+176)}:_{m} 2·HS:2 in characteristic other than 2 or 5 is indeed 133, the two possibilities of this degree giving rise to representations of the Harada–Norton group HN. Our definition of a true representation is motivated by our notion of a true image of a progenitor; thus we define a true representation of a progenitor T^{⋆n}:N to be a representation ρ for which Nρ ≅ N, T_{0}ρ ≅ T_{0} and all the T_{i}ρ are distinct. But in characteristic other than 2 or 5 the other progenitor 5^{⋆(176+176)}:_{m} 2·HS:2 also has two true representations of degree 133 (with the minimum being 133 also); these representations are not representations of HN.

In characteristic 5 the possible representations of the two progenitors 5^{⋆(176+176)}:_{m} 2·HS:2 are markedly different. Our progenitor has true representations of degree 57 [including irreducible ones]; the other progenitor does not have true representations of degree 57, or in smaller degrees.

7.1 Remarks on the representation

Presently, our representation of HN is written over ℤ[b_{5}, \frac{1}{2}, \frac{1}{5}], where b_{5} denotes the irrationality \frac{1}{2}(−1 + \sqrt{5}). Since ℤ[b_{5}] is a Euclidean domain, it is possible to convert our representation into a ℤ[b_{5}]-integral representation of HN. We have not done this, and the ‘integer explosion’ that would almost certainly result has made us reluctant to try.

It is impossible to eliminate the \frac{1}{2}’s from the representation without destroying the block diagonal form of N ≅ 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 we can eliminate \frac{1}{5}’s from the representation while still preserving the block diagonal form of 2·HS:2. We have not tried very hard to do this, but the main sticking point appears to be in finding a ℤ[b_{5}]-integral representation of HS:2 of type 77a whose 5-modular reduction has the required structure, namely 1 ⊕ 21 ⊕ 55.

Some effort has been expended to make sure that certain key representations of subgroups of N were expressed with respect to a ‘nice’ basis. In particular, the irreducible 77dimensional representation of HS:2 and the irreducible 56-dimensional representation of (U_{3}(5) × 2)·2 (the latter needed when constructing 2·HS:2) have sparse matrices and require just the integers in {−2, −1, 0, 1, 2}. [This applies to any element of the groups, not just their generators.] At the moment, a typical entry of our HN-representation, 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?) 1000 in modulus; the generators themselves are much nicer.

References

[1] J. N. Bray. Symmetric Presentations of Sporadic Groups and Related Topics. PhD thesis, University of Birmingham (1997).

[2] J. N. Bray. Progenitors and amalgamated products. In preparation.

[3] J. J. Cannon et al. The Magma programming language (various versions up to Version 2.8). School of Mathematics and Statistics, University of Sydney (1993–2001).

[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson. An 𝔸𝕋𝕃𝔸𝕊 of Finite Groups. Oxford University Press (1985).

[5] R. T. Curtis. Symmetric presentations I: Introduction, with particular reference to the Mathieu groups M_{12} and M_{24}. In Proceedings of 1990 LMS Durham Conference “Groups, Combinatorics and Geometry” (eds. M. W. Liebeck and J. Saxl), 380–396. LMS Lecture Note Series 165, Cambridge University Press, 1992.

[6] R. T. Curtis. Symmetric generation of the Higman–Sims group. J. Algebra 171 (1995), 567–586.

[7] R. T. Curtis. Monomial modular representations and construction of the Held group. J. Algebra 184 (1996), 1205–1227.

[8] A. Jabbar. Symmetric Presentations of Subgroups of the Conway Groups and Related Topics. PhD thesis, University of Birmingham (1992).

[9] S. P. Norton. F and other Simple Groups. PhD thesis, University of Cambridge (1975).

[10] S. Stanley. Monomial Representations and Symmetric Presentations. PhD thesis, University of Birmingham (1998).

[11] J. A. Todd. A note on the linear factional group. J. London Math. Soc. 7 (1932), 195–200.

Construction of HN 21

Table 2: Some characters of 2·HS (taken from the Atlas [4])

44352000 7680 2880 360 3840 256 64 500 300 25 36 24 7 16 16 16 20 20 11 11 12 15 20 20
p power  A  A A  A A A A A A AB AA A B C C AA BB A A BA BA AA AA
p′ part  A  A A  A A A A A A AB AA A A A A AA BB A A AA BA AA AA
ind 1A 2A 2B 3A 4A 4B 4C 5A 5B 5C 6A 6B 7A 8A 8B 8C 10A 10B 11A B∗∗ 12A 15A 20A B∗∗
+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
+ 22  6 −2 4 −6 2 2 −3 2 2 −2 0 1 0 0 0 1 −2 0 0 0 −1 −1 −1
+ 77 13 1 5 5 5 1 2 −3 2 1 1 0 1 −1 −1 −2 1 0 0 −1 0 0 0
+ 154 10 10 1 −2 6 −2 4 4 −1 1 1 0 0 0 0 0 0 0 0 1 1 −2 −2
+ 154 10 −10 1 −10 −2 2 4 4 −1 −1 1 0 0 2 −2 0 0 0 0 −1 1 0 0
+ 154 10 −10 1 −10 −2 2 4 4 −1 −1 1 0 0 −2 2 0 0 0 0 −1 1 0 0
+ 175 15 11 4 15 −1 3 0 5 0 2 0 0 −1 1 1 0 1 −1 −1 0 −1 0 0
+ 231  7 −9 6 15 −1 −1 6 1 1 0 −2 0 −1 −1 −1 2 1 0 0 0 1 0 0
ind  1  2  4 3  4 4 4 5 5 5 12 6 7 8 8 8 10 20 11 11 12 15 20 20
2  2  6  4  10 10 10 12 6 14 8  10  22 22 12 30 20 20
+ 56  8  0 2  0 0 0 6 −4 1 0 2 0 0 0 0 −2 0 1 1 0 2 0 0
∘ 176 16  0 5 16i 0 0 1 6 1 3i 1 1 0 0 0 1 0 0 0 −i 0 i i
∘ 176 16  0 5 −16i 0 0 1 6 1 −3i 1 1 0 0 0 1 0 0 0 i 0 −i −i