The University of Southampton
Courses

# MATH6156 Modules and Representations

## Module Overview

This module provides an introduction to the theory of modules over a principal ideal domain and the representation theory of finite groups, two basic tools in advanced mathematics.

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Understand the notion of a module over a ring, and the basic properties of a free module;
• Understand and apply the classification of finitely generated modules over a principal ideal domain.
• understand the definitions and basic properties of the classical linear groups;
• calculate the irreducible representations and character tables of some small finite groups

### Syllabus

• Modules: definitions, first examples; basic properties; submodules; factor modules; isomorphism theorems; correspondence theorem. • Free modules; rank; universal property; free modules over integral domains; the torsion submodule. • Modules over a principal ideal domain; The classification of finitely generated modules over a principal ideal domain. • The classification of finitely generated abelian groups. • The Jordan normal form of matrices over the complex numbers. • Matrix groups; general and special linear, orthogonal, symplectic and unitary groups; possibly a survey of crystallographic groups in dimensions 2 and 3. • Representation theory for finite groups over the complex numbers; Schur’s Lemma, Maschke’s Theorem, character theory and examples of character tables in small examples. • Burnside’s p-q Theorem

### Learning and Teaching

#### Teaching and learning methods

Lectures, printed notes, private study

TypeHours
Revision30
Lecture36
Preparation for scheduled sessions24
Follow-up work24
Total study time150

ALPERIN J L & BELL R B. Groups and representations.

CAMERON, P J. Introduction to algebra.

ELLIOTT J P & DAWBER P G. Symmetry in Physics, vol. 1.

LANG S. Algebra.

CURTIS M L. Matrix Groups.

FULTON W & HARRIS J,. Representation Theory.

GORDON, J & Liebeck, M. Representations and Characters of Finite Groups.

SERRE J-P. Linear Representations of Finite Groups.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 50%
Written assessment 50%

#### Referral

MethodPercentage contribution
Written assessment 100%

#### Repeat Information

Repeat type: Internal & External

Pre-requisite: MATH3086

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Recommended texts for this module may be available in limited supply in the University Library and students may wish to purchase reading texts as appropriate.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.