Intensive course Permutation groups

Lecturer

Rick Thomas (Leicester)

Schedule

This intensive course will be held 4 June - 13 June 2003 (Wed-Sat, Mon-Fri) at the University of Helsinki in the Department of Mathematics.

Time Lecture room
Wednesday 4.6.12-14 SIII
Thursday 5.6. 12-14 SIII
Friday 6.6. 10-13 SIII
Saturday 7.6. 10-12 SIII
Monday 9.6. 10-13 SIV
Tuesday 10.6. 10-12 SIV
Wednesday 11.6. 10-12 SIV
Thursday 12.6. 10-12 SIV
Friday 13.6. 10-12 SIV

Outline of course

The purpose of this course is to give an introduction to some topics in the theory of permutation groups.

We do not presume much in the way of assumed knowledge: essentially just some elementary group theory (such as the notion of a normal subgroup). We will cover the pre-requisites at the start of the course. There will also be a survey of the relevant properties of the alternating and symmetric groups.

After this introduction, we will then cover the basics of permutation groups, such as the notions of permutation representations, orbits and stabilizers, and we will show how these ideas allow us to prove results in group theory (such as Sylow's theorems). We will also cover Burnside's lemma.

Following on, we look at the notions of multiple transitivity and primitivity, and demonstrate properties of groups having permutation representations of this kind (such as properties of normal subgroups). We will also give an indication of more advanced topics, such as Jordan groups and the O'Nan-Scott theorem.

Throughout the course, we will be giving examples showing how these notions relate to other areas of mathematics (such as graph theory).

Course material

Lecture notes

Context

This intensive course is one in the series of algebra courses arranged in connection to the Finite Model Theory project:

Department of Mathematics Finite model theory in Finland Teaching of mathematical logic