Permutation Groups notes and preprints

• Combinatorics Study Group papers
• Lecture notes in design theory and combinatorics
• Lecture notes:
  • R. A. Bailey, Association schemes
  • John Beachy, Abstract algebra on-line
  • A. Betten, H. Fripertinger and A. Kerber, Algebraic combinatorics via finite group actions
  • Peter J. Cameron, Classical groups
  • Peter J. Cameron, Polynomial aspects of codes, matroids and permutation groups
  • The Dog School's Introduction to Group Theory
  • J. W. P. Hirschfeld, Semi-linear groups over finite fields
  • W. D. Joyner, The mathematics of Rubik's cube
  • O. H. King, Classical groups
  • M. Klin et al., Algebraic Combinatorics in Mathematical Chemistry
  • J. S. Milne's Course Notes
  • T. W. Müller, Five lectures on generalized permutation representations
  • CSUSB Mathematics Reference Notes

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)

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
http://www.math.niu.edu/~beachy/aaol/
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
Contents:

• Format: Index of definitions; index of theorems; list of theorems.
• Content: Integers, functions, groups, polynomials, rings, fields, Galois theory, modules

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
Contents:

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
http://www.maths.qmul.ac.uk/~pjc/class_gps/.
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
Contents:

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
http://www.maths.qmul.ac.uk/~pjc/csgnotes/cmpgpoly.pdf.
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
Contents:

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
http://members.tripod.com/~dogschool/
Learn group theory "as a dog learns" (Shaw, Caesar and Cleopatra), and end up surpassing the philosophers. Nothing superfluous here!
Format: HTML
Contents:

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
http://dwispc8.vub.ac.be/Potenza/lectnotes.html
Discusses the types of polarities of projective spaces and the semi-linear groups they define.
Format: PostScript
Contents:

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
http://www.permutationpuzzles.org/rubik/webnotes/
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
Contents:

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

O. H. King, Classical groups, (Socrates course notes), available from
http://dwispc8.vub.ac.be/Potenza/lectnotes.html
A general account of classical groups including Aschbacher's Theorem.
Format: PostScript
Contents:

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
http://www-lit.ma.tum.de/veroeff/html/950.05003.html
This valuable exposition and survey brings together ideas about graph isomorphism, cellular algebras, permutation groups, and mathematical chemistry.
Format: PostScript
Contents:

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
http://www.jmilne.org/math/CourseNotes/index.html
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
Contents:

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

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(G, H wr S_n) and subgroup growth

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.