General description
Ever since the basic work of Frobenius (1896) and Schur (1901), representing groups by groups of matrices has been regarded as an invaluable tool in the study of groups. In more modern times there have been important applications not only in other branches of mathematics but also in physics and chemistry. This module contains all the background ideas with an emphasis on matrix representations and their characters and also on explicit methods to calculate these in important particular examples such as symmetric groups.

Aims
To introduce students to the concepts and the general theory of the representation of groups by matrices, to familiarise students with practical methods for constructing representations and character tables of groups and to enable them to obtain information about the structure of groups from their representations and character tables.

Objectives
A student who successfully completes this course should be able to:

• describe the concepts of irreducibility and equivalence of representations;
• calculate permutation representations of groups;
• demonstrate the value of the theory of group characters in obtaining information about the structure of particular groups;
• prove basic results in representation theory, eg Maschke's Theorem, Schur's Lemma and theorems involving the concepts of equivalence and irreducibility;
• calculate character tables of groups of small order;
• calculate character tables of symmetric groups and their subgroups;
• determine irreducible representations of cyclic, dihedral and symmetric groups.

Syllabus
1. MATRIX REPRESENTATION OF GROUPS
Examples for cyclic groups, symmetric groups, permutation groups. Permutation and regular representations. Equivalence of representations.

2. IRREDUCIBLE REPRESENTATIONS
Reducible and completely reducible representations. Mashke's Theorem.

3. GROUPS CHARACTERS
Permutation and regular character. Examples. Condition of irreducibility and equivalence in terms of characters. Number of irreducible characters equal to number of classes of conjugate elements. Character tables. Schur's lemma and relations. Orthogonality relations.

4. CONSTRUCTION OF REPRESENTATIONS
Direct product of representations. Induced representations and characters. Lifting of representations.

5. CALCULATION OF CHARACTER TABLES
Programme for claculating character tables. Many examples, cyclic groups, abelian groups, metacyclic groups, symmetric and alternating groups.

6. APPLICATIONS TO GROUP THEORY
Normal subgroups. Simple groups. Centres of groups. Burnside's Theorem.

7. PROJECTIVE REPRESENTATIONS OF GROUPS