# MA3E1 Content

Content: The concept of a group is defined abstractly (as set with an associative binary operation, a neutral element, and a unary operation of inversion) but is better understood through concrete examples, for instance:

• Permutation groups
• Matrix groups
• Groups defined by generators and relations.

All these concrete forms can be investigated with computers. In this module we will study groups by:

• Finding matrix groups to represent them
• Using matrix arithmetic to uncover new properties. In particular, we will study the irreducible characters of a group and the square table of complex numbers they define. Character tables have a tightly-constrained structure and contain a great deal of information about a group in condensed form. The emphasis of this module will be on the interplay of theory with calculation and examples.

Aims: To introduce representation theory of finite groups in a hands-on fashion.

Objectives: To enable students to:

• Understand matrix and linear representations of groups and their associated modules
• Compute representations and character tables of groups
• Know the statements and understand the proofs of theorems about groups and representations covered in this module.

Books:

Many books explain the theory well and give different examples:

James and Liebeck - Representations and Characters of Groups;

Steinberg - Representation Theory of Finite Groups; An Introductory Approach;

Sagan - The Symmetric Group; Representations, Combinatorial Algorithms and Symmetric Functions;

Fulton and Harris - Representation Theory; A First Course;

Serre - Linear Representations of Finite Groups.