Introductory description

See outline syllabus.

Module web page

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.

The main emphasis of this course will be on finite groups, and the classification of groups of small order. However, results will be stated for infinite groups too whenever possible.

• Permutation groups and groups acting on sets. The Orbit-Stabiliser Theorem. Conjugacy Classes.
• The Sylow Theorems. Direct and semidirect products of groups.
• Classification of groups of order up to 20 (except 16).
• Nilpotent and soluble groups.
• More on permutation groups. Primitivity and multiple transitivity.
• Groups of matrices. Simplicity of the alternating groups and the groups PSL(n,K).
• The transfer homomorphism. Burnside's transfer theorem.
• Classification of finite simple groups of order up to 500.

Learning outcomes

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

• By the end of the module students should be familiar with the topics. In particular, they should be able to prove Sylow's Theorems, and to use them and other techniques as a tool for analysing the structure of a finite group of a given order.

Subject specific skills

The group theory module offers an introduction to the classification of finite simple groups, one of the handful of the most significant mathematical achievements of the late twentieth century. Students taking the module learn some of the techniques required for working on a large-scale research project. These techniques are partly algorithmic and computational, and include the development of databases for storing and retrieving information
about specific groups.

Transferable skills

Clear and precise thinking, and the ability to follow complex reasoning, to construct logical arguments, and to expose illogical ones. The ability to retrieve the essential details from a complex situation and thereby facilitate
problem resolution. Formulating problems as algorithms, thereby enhancing understanding of details and rendering them suitable for computer implementation.