# MA442 Group Theory

**Lecturer: **Derek Holt

**Term:** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 lectures

**Assessment:** Three-hour written examination (100%)

**Formal registration prerequisites: **None

**Assumed knowledge: ** 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.

We start with a review of basic material on permutation groups and group actions followed by Sylow's theorems and groups of small order. This was (almost) all covered in MA3K4.

The topic of groups defined by presentations using generators and defining relations (which are used in algebraic topology for example) will be covered more formally than in earlier group theory courses, with plenty of examples. These groups are defined as quotient groups of free groups, so we shall start with the definition and basic properties of free groups. We shall also introduce the Todd-Coxeter coset enumeration procedure, again with lots of examples. This method can often be used to prove that a given presentation defines a finite group.

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.

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 200.

**Aims:** The two principal aims of the module are to introduce the formal theory of group presentations together with the coset enumeration algorithm, and to classify all simple groups of order up to 200. 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 apply the coset enumeration algorithm to help with the analysis of groups defined by a presentation, and they should know how 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.

One important difference in notation will be that we shall use right actions rather than left actions, which were used in MA3K4.