Queen Mary, University of London
School of Mathematical Sciences

Permutation Groups notes and preprints

Note on formats: HTML files should be handled by your browser. Others require special software to display or print them. On my system, these are xdvi (for DVI), ghostview (for PostScript), and acroread (Acrobat reader for PDF). All these are freely available and can be wired into your browser.

Combinatorics Study Group Papers

Available from
An occasional series of expository papers about topics discussed in the Study Group.
  1. Problems from the First Anglo-Hungarian Meeting on Groups and Geometries, in DVI or PostScript format.
  2. Finitary permutation groups, by Chris Pinnock (PDF format)
  3. Five lectures on generalized permutation representations, by Thomas Müller (PostScript) - see below.
  4. Borcherds' proof of the moonshine conjecture, after V. Nikulin (PDF format)
  5. Partially ordered sets, by Thomas Britz and Peter Cameron (PDF format)
Other notes (not particularly about permutation groups) are also available.

Lecture notes

R. A. Bailey, Association schemes, available from
Notes of a course currently in progress. Written from a statistician's point of view, these notes are complementary to the treatments by Bannai and Ito or by Brouwer, Cohen and Neumaier: they have much to say about methods of calculation and about highly imprimitive association schemes, for example. Also includes an annotated reading list, and course problem sheets. The notes are adapted from a forthcoming book.
Format: PDF
  1. Definitions of association scheme
  2. Adjacency matrices
  3. Some special association schemes
  4. The Bose-Mesner algebra
  5. Character tables
  6. Techniques
  7. Strongly regular graphs
  8. Block designs
  9. Partially balanced block designs
  10. A little statistics
  11. Efficiency
  12. Cyclic Designs
  13. Families of Partitions
  14. Orthogonal block structures

John Beachy, Abstract algebra on-line, available from
Not notes, but an extensive resource for undergraduate students containing all the important definitions and theorems from a wide cross-section of abstract algebra.
Format: HTML

A. Betten, H. Fripertinger and A. Kerber, Algebraic combinatorics via finite group actions, available from http://www.mathe2.uni-bayreuth.de/frib/html/book/hyl00.html
A very complete survey of enumeration under group action; in interactive HTML, so you can try out the concepts for yourself. Many exercises. But you have to re-configure your X-windows to get the symbols to print correctly.
Format: HTML

  1. Actions (Actions of groups; Bilateral classes, symmetry classes of mappings; Finite symmetric groups; Complete monomial groups; Enumeration of symmetry classes; The involution principle; Special symmetry classes)
  2. Weights (Enumeration by weight; Cycle indicator polynomials; Sums of cycle indicators, recursive methods; A generalization; The Decomposition Theorem; Species)
  3. Marks
  4. Constructions (Orbit evaluation; Transversals of symmetry classes; Orbits of centralizers; Recursion and orderly generation; Generating orbit representatives; Symmetry adapted bases)
  5. Index

Peter J. Cameron, Classical groups, available from
Notes of a lecture course, roughly following Taylor's book: generation and simplicity of classical groups, and some of their geometry. Includes exercises.
Format: DVI, PostScript, PDF

  1. Fields and vector spaces
  2. Linear and projective groups
  3. Polarities and forms
  4. Symplectic groups
  5. Unitary groups
  6. Orthogonal groups
  7. The Klein correspondence and triality
  8. Further topics
  9. A short bibliography on classical groups

Peter J. Cameron, Polynomial aspects of codes, matroids and permutation groups, available from
These notes include background on codes, matroids and permutation groups, and polynomials associated with them (weight enumerator, Tutte polynomial and cycle index), and describe the links between these objects. Their second purpose is to describe codes over Z4 and the associated matroids and permutation groups.
Format: PDF

  1. Codes
  2. Codes over Z4
  3. Matroids
  4. Matroids and codes
  5. Permutation groups
  6. Cycle index
  7. Codes and permutation groups
  8. IBIS groups

The Dog School of Mathematics, Introduction to Group Theory, available from
Learn group theory "as a dog learns" (Shaw, Caesar and Cleopatra), and end up surpassing the philosophers. Nothing superfluous here!
Format: HTML

  1. What is Group Theory?
  2. Examples of Groups
  3. Housekeeping Theorems
  4. Cayley Tables
  5. Symmetry Group of the Triangle
  6. Subgroups
  7. Cosets
  8. Lagrange's Theorem
  9. Cyclic Groups and Subgroups
  10. Permutations
  11. Permutation Groups
  12. Rubik's Cube
  13. Rubik's Cube Groups
  14. Solve the Cube 1

J. W. P. Hirschfeld, Semi-linear groups over finite fields, (Socrates course notes), available from
Discusses the types of polarities of projective spaces and the semi-linear groups they define.
Format: PostScript

  1. Polarities
  2. Groups on the line
  3. Orders and isomorphisms among the semi-linear groups
  4. Bibliography

W. D. Joyner, The mathematics of Rubik's cube, available from
An introduction to the discrete mathematics and group theory underlying Rubik's cube and other permutation puzzles. A fine example of non-trivial mathematics arising from "diversions". Many worked examples and exercises.
Format: DVI

  1. Logic and sets
  2. Functions, matrices, relations and counting
  3. Permutations
  4. Permutation puzzles
  5. Groups, I
  6. Graphs and "God's Algorithm"
  7. Symmetry groups of the Platonic solids
  8. Groups, II
  9. The Rubik's cube and the word problem
  10. The 2 × 2 and 3 × 3 cube groups
  11. Other Rubik-like puzzle groups
  12. Interesting subgroups of the cube group
  13. Crossing the Rubicon
  14. Appendix: some solution strategies
Note added 3 June 2002: The dvi file is no longer available; however, there are compressed PostScript and PDF versions at http://www.mic.atr.co.jp/~gulliver/Rubik/. (Thanks to Lewis Nowitz for this information.)

O. H. King, Classical groups, (Socrates course notes), available from
A general account of classical groups including Aschbacher's Theorem.
Format: PostScript

  1. Forms and groups
  2. Isomorphisms between classical groups
  3. Aschbacher's Theorem
  4. Bibliography

M. Klin, Ch. Rücker, G. Rücker, G. Tinhofer, Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms, I, Permutation Groups and Coherent (Cellular) Algebras, available from
This valuable exposition and survey brings together ideas about graph isomorphism, cellular algebras, permutation groups, and mathematical chemistry.
Format: PostScript

  1. Introduction
  2. The subject of algebraic combinatorics
  3. Problems related to the perception of the symmetry of chemical graphs
  4. Fundamentals of permutation group theory
  5. Centralizer algebras of permutation groups
  6. Cellular algebras
  7. Galois correspondence between permutation groups and cellular algebras
  8. S-rings over cyclic groups
  9. Automorphism groups of certain chemical graphs
  10. Concluding remarks

J. S. Milne's course notes, available from
Full notes of all the advanced courses the author has taught since 1986, available in a single file or in separate sets of notes. Plenty of food for thought here.
Format: DVI, HTML

  1. Group Theory
  2. Fields and Galois Theory
  3. Algebraic Number Theory
  4. Class Field Theory
  5. Modular Functions and Modular Forms
  6. Elliptic Curves
  7. Algebraic Geometry
  8. Lectures on Etale Cohomology
  9. Abelian Varieties
T. W. Müller, Five lectures on generalized permutation representations, available from
Lectures given by the author in an algebra summer school in Brazil, and repeated in the Queen Mary Combinatorics Study Group. They describe techniques for counting representations of an arbitrary finitely generated group in a wreath product H wr Sn (or a variant on this), with applications to such topics as Quillen complexes and subgroup growth.
Format: PostScript
  1. Some combinatorial aspects of permutation representations
  2. Generalizing permutation representations
  3. Some examples and a formula for the exterior function
  4. Explicit formulae for abelian groups and computations in Quillen complexes
  5. Asymptotics of Hom(GH wr Sn) and subgroup growth
CSUSB Mathematics Reference Notes, available from
Foundational material on logic, functions, relations, and group theory. Index, search facilities, and web tools provided.
Format: HTML
  1. Notes in set theory
  2. Notes in symbolic logic
  3. Notes on methods of proof
  4. Notes on basic proofs
  5. Notes on functions
  6. Notes on relations
  7. Notes on binary operations
  8. Notes on groups

Peter J. Cameron
8 May 2002.

Permutation Groups Pages at Queen Mary:
Resources | Lecture Notes | Problems | Exercises | The Book | Problems from "Permutations"
Maths Research Centre