# MA3E1 Groups & Representations

**Lecturer:** Samir Siksek

**Term(s):** Term 1

**Status for Mathematics students:** List A

**Commitment:** 30 one-hour lectures

**Assessment:** Homework 15%, 3 hour written exam 85%

**Formal registration prerequisites: **None

**Assumed knowledge: **

- MA132 Foundations or MA138 Sets and Numbers
- MA136 Introduction to Abstract Algebra
- MA106 Linear Algebra
- MA251 Algebra I: Advanced Linear Algebra
- MA249 Algebra II: Groups and Rings

**Useful background:** Only the above

**Synergies: **This module goes well with other third year algebra modules, particularly Group Theory.

**Leads to: **The following modules have this module listed as **assumed knowledge** or **useful background:**

**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
- 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).