MA4P3 Classical Groups

Introductory description

Classical groups are a fundamental class of matrix groups that appear throughout mathematics and physics, often described as groups of automorphisms that preserve a bilinear or sesquilinear form on a finite-dimensional vector space. They are closely related to the geometry of Euclidean spaces and are central to the classification of finite simple groups; the latter are the building blocks of finite and infinite groups. This module provides an introduction to the subject of classical groups that does not assume knowledge beyond the second year algebra.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Introduction and revision of group actions.
   1. General and special linear groups.
   2. Introduction to projective geometry.
   3. Sesquilinear forms.
   4. Classical groups and their simplicity.
   5. Geometric subgroups of classical groups.

Learning outcomes

By the end of the module, students should be able to:

• Define the classical groups over arbitrary fields.
• Establish the simplicity of many of the classical groups.
• Understand important aspects of the subgroup structure of the classical groups, using the underlying geometry.

Indicative reading list

Reading lists can be found in Talis

Subject specific skills

Develop a deep understanding and applicability of the following topics:
◦ Simplicity of certain classical groups;  
◦ The underlying geometry of the classical groups;
◦ Constructing geometric subgroups of classical groups;
◦ Finding generating sets for classical groups using geometric techniques.

Transferable skills

Be able to transfer knowledge between distinct areas of mathematics (in this case Algebra and Geometry) to prove results in both areas.