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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
Lecture36
Completion of assessment task24
Preparation for scheduled sessions24
Follow-up work24
Wider reading or practice12
Revision30
Total study time150

Resources & Reading list

CURTIS M L. Matrix Groups. 

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

CAMERON, P J. Introduction to algebra. 

SERRE J-P. Linear Representations of Finite Groups. 

FULTON W & HARRIS J,. Representation Theory. 

LANG S. Algebra. 

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

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

Assessment

Summative

MethodPercentage contribution
Coursework 20%
Exam 80%

Referral

MethodPercentage contribution
Exam 100%

Repeat Information

Repeat type: Internal & External

Linked modules

Pre-requisite: MATH3086

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