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 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;
- calculate the irreducible representations and character tables of some small finite groups
- Understand and apply the classification of finitely generated modules over a principal ideal domain.
- 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
|Preparation for scheduled sessions||24|
|Completion of assessment task||24|
|Wider reading or practice||12|
|Total study time||150|
Resources & Reading list
SERRE J-P. Linear Representations of Finite Groups. Springer.
ALPERIN J L & BELL R B. Groups and representations. Springer.
GORDON, J & Liebeck, M. Representations and Characters of Finite Groups.
LANG S. Algebra. Springer.
ELLIOTT J P & DAWBER P G. Symmetry in Physics, vol. 1. MacMillan.
CURTIS M L. Matrix Groups. Springer.
CAMERON, P J. Introduction to algebra. OUP.
FULTON W & HARRIS J,. Representation Theory. Springer.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External