# MA3E1 Groups & Representations

**Lecturer:** Miles Reid

**Term(s):** Term 2

**Status for Mathematics students:** List A

**Commitment:** 30 one-hour lectures

**Assessment:** Assigned work/tests 15%. Three-hour written exam 85%

**Prerequisites:** The Group theory and linear algebra taught in core modules

**Leads To: **

**Content**: The concept of a group is defined abstractly (as set with an associative binary operation, a neutral element, and a unary operation of inversion) but is better understood through concrete examples, for instance

- permutation groups
- matrix groups
- groups defined by generators and relations. All these concrete forms can be investigated with computers. In this module we will study groups by
- finding matrix groups to represent them
- using matrix arithmetic to uncover new properties. In particular, we will study the irreducible characters of a group and the square table of complex numbers they define. Character tables have a tightly-constrained structure and contain a great deal of information about a group in condensed form.
*The emphasis of this module will be on the interplay of theory with calculation and examples.*

**Aims**: To introduce representation theory of finite groups in a hands-on fashion.

**Objectives**: To enable students to:

- understand matrix and linear representations of groups and their associated modules,
- compute representations and character tables of groups, and
- know the statements and understand the proofs of theorems about groups and representations covered in this module.

**Books**:

We will work through printed notes written by the lecturer.

A nice book that we shall not use is:

G James & M Liebeck, *Representations and Characters of Groups*, Cambridge University Press, 1993. Second edition, 2001. (IBSN: 052100392X).