Lecturer: Derek Holt
Term: Term 2
Status for Mathematics students: List C
Commitment: 30 lectures
Assessment: Three-hour written examination (100%)
Prerequisites: MA3K4 Introduction to Group Theory
Content: This module is a continuation of MA3K4 Introduction to Group Theory. The material in MA3K4 will be reviewed as we go through, but it will be assumed that students are already familiar with it. One significant difference in notation is that we shall be using right rather than left actions.
The main emphasis will be on finite groups, and particularly on the classification of simple groups of small order. However, results will be stated for infinite groups too whenever possible.
After reviewing the basic material on permutation groups and group actions, we shall study the Schreier-Sims Algorithm, which is an efficient programmable algorithm for computing the order of a subgroup of the symmetric group Sym(n) generated by a given set of permutations. This is based on and uses the Orbit-Stabilizer Theorem, together with a general theorem of Schreier on generating sets of subgroups of groups.
Soluble groups were studied in MA3K4, and we shall introduce and develop the basic properties of the more restricted class of nilpotent groups, which includes all finite $p$-groups.
Transitive and doubly transitive groups were introduced in MA3K4, and we shall study the intermediate classes of primitive and imprimitive permutation groups. Imprimitive groups arise naturally when the set X of permuted elements can be partitioned into sets of more than one element that are permuted by the group. For example, the Rubik's Cube group acting on the 24 corner faces of the cube permutes the eight corners, each of which consists of three faces. This leads to a new general construction, the wreath product of two groups, which is based on both direct and semidirect products.
The final part of the module will be on finite simple groups. We shall review the proof (from MA3K4) of the simplicity of the alternating groups Alt(n) for n at least 5, and prove the simplicity of the groups PSL(n,K) (which arise as quotient groups of groups of matrices over a field K) for all n greater than 1 (with a couple of small exceptions).
We finish with a complete classification of simple groups of order up to 500.
Aims: The main aim of this module is to classify all simple groups of order up to 500. The module will give some of the flavour of the greatest achievement in group theory during the 20th century, namely the classification of all finite simple groups.
Objectives: By the end of the module students should be familiar with the topics listed above under `Contents'. In particular, they should be able to use Sylow's Theorems and other techniques as a tool for analysing the structure of a finite group of a given order.
Books: No specific books are recommended for this module. There are many groups on Group Theory in the library, and some of these might be helpful for parts of the module, but no single book is likely to cover the whole syllabus.