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
Having successfully completed this module you will be able to:
- Understand and apply the classification of finitely generated modules over a principal ideal domain.
- calculate the irreducible representations and character tables of some small finite groups
- Understand the notion of a module over a ring, and the basic properties of a free module;
- understand the definitions and basic properties of the classical linear groups;
- 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
|Wider reading or practice||12|
|Preparation for scheduled sessions||24|
|Completion of assessment task||24|
|Total study time||150|
Resources & Reading list
ALPERIN J L & BELL R B. Groups and representations. Springer.
ELLIOTT J P & DAWBER P G. Symmetry in Physics, vol. 1. MacMillan.
CURTIS M L. Matrix Groups. Springer.
GORDON, J & Liebeck, M. Representations and Characters of Finite Groups.
LANG S. Algebra. Springer.
SERRE J-P. Linear Representations of Finite Groups. Springer.
FULTON W & HARRIS J,. Representation Theory. Springer.
CAMERON, P J. Introduction to algebra. OUP.
Summative assessment description
Referral assessment description
Repeat type: Internal & External