Permutation Groups

This Web page is associated with the book Permutation Groups, by Peter J. Cameron, published by Cambridge University Press in the London Mathematical Society Student Texts series.

Here you will find, in addition to notes and links:

the contents and preface in PostScript;
a list of misprints and corrections;
further exercises;
solutions to the Exercises in the book (someday...);
a note on Burnside's Theorem (Theorem 4.3);
a terminological note on cellular algebras;
commented GAP code for the examples in Chapter 1 and Chapter 3. The code for Chapter 3 now constructs the 2-transitive action of the Higman-Sims group.
a short note about permutation characters;
update to the bibliography.

Notes

1. Lewis Nowitz points out that in Figure 3.3, the cube on the right, which is used as an aid to explain the concept of antipodal equivalence classes, is labelled clockwise in bridge-suit order: clubs, diamonds, hearts, and spades. Therefore, although the cube can't be duplicated, it can be doubled...

2. As promised, the GAP code in the book has been checked on GAP4 and works correctly, though the output is not in quite the same form as in the book; sometimes more or less detail is given. Two new features of GAP4 are worth pointing out. There is now a command AutomorphismGroup which can be applied to either a group or a graph, and a command Transitivity which gives the degree of transitivity of a permutation group. I have used these in the commented GAP code for Chapter 3.

3. The reference to the classification of the affine 2-transitive groups (p.110) is inadequate and should be supplemented with the following papers:

Cristoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II. J. Algebra 93 (1985), 151-164.
Martin W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. (3) 54 (1987), 477-516.

4. Reviews:

From the review by W. Knapp in Zentralblatt für Mathematik:

... an excellent concise account of the modern theory of permutation groups including many recent developments... In spirit and scope it may be considered as a (post-)modern equivalent of H. Wielandt's famous book "Finite Permutation Groups" ... A special feature is given by the worked examples using the computer algebra system GAP.

From the review in the EMS Newsletter:

Students should find this book very stimulating because of the many different connections it mentions ...

Links

Permutation Groups resources (under construction)
Euler Institute for Discrete Mathematics and Applications (you can see the original course announcement here)
Design Research Group homepage
GAP homepage (including the manuals for GAP 3 and GAP 4)
Brouwer's DRG finder (READ THE INSTRUCTIONS BELOW)
Alexander Hulpke's transitive groups of small degree
Matroids page
ATLAS of Group Representations
Permutation group problems
Proof of Bertrand's Postulate by Robin Chapman
"Permutations", preprint of paper for the Erdös memorial conference (dvi or PostScript). You can find an update on the problems from this paper here.
A course on Permutation groups, structures and polynomials at Charles University, Prague, in 2004.

Instructions for Brouwer's DRG finder
The Subject of your mail must be

        exec drg

The body of the mail should contain any number of lines of the form

        drg d=D B[0],B[1],...,B[d-1]:C[1],...,C[d]

where D is the diameter and B[i], C[i] the standard parameters.

Updates to references

References [9], [12], [48], [81], [119], [131] and [165] have all appeared:
[9] László Babai and Peter J. Cameron, Automorphisms and enumeration of switching classes of tournaments, Electronic J. Combinatorics 7 (2000), #38 (25pp.)
[12] R. A. Bailey, Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2004.
[48] Peter J. Cameron and Csaba Szabó, Independence algebras, J. London Math. Soc. (2) 61 (2000), 321-334.
[81] L. A. Goldberg and M. R. Jerrum, The "Burnside process' converges slowly, in: Proceedings of Random 1998, Randomisation and Approximation Techniques in Computer Science, Lecture Notes in Computer Science 1518, Springer-Verlag, pp. 331-345.
[119] Martin W. Liebeck and Aner Shalev, Simple groups, permutation groups, and probability, J. Amer. Math. Soc. 12 (1999), 497-520.
[131] Dugald Macpherson, Sharply multiply homogeneous permutation groups, and rational scale types, Forum Math. 8 (1996), 501-507.
[165] Ákos Seress, Permutation Group Algorithms, Cambridge Tracts in Mathematics 152, Cambridge University Press, 2003.

Reference [17] is now more accessible; it has been published by Springer-Verlag as volume 12 in the series Texts and Readings in Mathematics in 1998.

A proof of the Strong Perfect Graph Conjecture (see p.155) has been announced by M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas.

Here are some further references:

P. J. Cameron, Permutations, pp. 205-239 in Paul Erdös and his Mathematics, Vol. II, Bolyai Society Mathematical Studies 11, Budapest, 2002.
Peter J. Cameron, Cycle index, weight enumerator and Tutte polynomial, Electronic J. Combinatorics 9(1) (2002), #N2 (10pp).
Peter J. Cameron, Michael Giudici, Gareth A. Jones, William M. Kantor, Mikhail H. Klin, Dragan Marusic and Lewis A. Nowitz, Transitive permutation groups without semiregular subgroups, J. London Math. Soc. (2) 66 (2002), 325-333.
Peter J. Cameron and C. Y. Ku, Intersecting families of permutations, Europ. J. Combinatorics 24 (2003), 881-890.
Clara Franchi, On permutation groups of finite type, European J. Combinatorics 22 (2001), 821-837.
Daniele A. Gewurz, Reconstruction of permutation groups from their Parker vectors, J. Group Theory 3 (2000), 271-276.
Michael Giudici, Quasiprimitive groups with no fixed point free elements of prime order, J. London Math. Soc. (2) 67 (2003), 73-84.
Martin W. Liebeck and Aner Shalev, Bases of primitive permutation groups, pp. 147-154 in Groups, Combinatorics and Geometry (ed. A. A. Ivanov, M. W. Liebeck and J. Saxl), World Scientific, New Jersey, 2003.
A. Maróti, On the orders of primitive groups, J. Algebra 258 (2002), 631-640.

Further references will be added here from time to time.

My homepage

Peter J. Cameron
27 August 2004.

Permutation Groups Pages at Queen Mary:

Resources | Lecture Notes | Problems | The Book | Problems from "Permutations"

Maths Research Centre