DocCourse, Prague, JanApr 2004
This page contains information about my lectures at the doctoral course on
Combinatorics, Geometry, and
Computation at the Charles University, Prague, January  March 2004.
For more information on the courses in the programme, and how to apply,
see the web page (link above).
Title
Permutation groups, structures, and polynomials
Abstract
The course will begin by discussing the basic concepts of permutation
groups, both finite and infinite. Many examples of infinite permutation
groups are obtained from homogeneous or omegacategorical structures, and
the course will describe some of these. Finite permutation groups occur
in enumeration theory; the cycle index of a permutation group has some
connections with the Tutte polynomial of a matroid. The theory of species
relates these two areas, studying an infinite structure by its finite subsets.
Problems
Course plan
 Week 1: Permutation groups
 Basic theory of permutation groups
 Orbitcounting and cycle index
 Primitivity, multiple transitivity
 Week 2: Finite permutation groups
 Primitive groups
 O'NanScott theorem
 Applications of CFSG
 Week 3: Homogeneous structures
 The random graph
 The ERS and Fraïssé theorems
 Other homogeneous structures
 Week 4: Counting finite substructures
 Species and their cycle index
 Product, substitution, derivative
 qanalogues
 Week 5: Polynomials and structures
 Codes and matroids
 Weight enumerator and Tutte polynomial
 Cycle index and Tutte polynomial
 Week 6: Revision and exam
Reading
Much of the course material can be found in the following sources:

P. J. Cameron, Permutation Groups,
LMS Student Texts 45, Cambridge University Press, Cambridge, 1999.

P. J. Cameron, The random graph, pp. 331351 in The Mathematics of
Paul Erdös (ed. R. L. Graham and J. Nesetril), Springer,
Berlin, 1997.

P. J. Cameron, Polynomial aspects of codes, matroids and permutation groups,
http://www.maths.qmw.ac.uk/~pjc/csgnotes/cmpgpoly.pdf
Background on matroids and species is not specifically required (all that is
needed will be covered in the lectures) but can be found in

J. G. Oxley, Matroid Theory, Oxford University Press, Oxford, 1992.

F. Bergeron, G. Labelle and P. Leroux,
Combinatorial Species and Treelike Structures,
Encyclopedia of Mathematics and its Applications 67,
Cambridge University Press, Cambridge, 1998.
Peter J. Cameron (P.J.Cameron@qmul.ac.uk)
2 March 2004