Content: 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. (Much of this material will have been covered already in MA249.)
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.
Aims: The main aim of this module is to classify all simple groups of order up to 500. Techniques will include the theorems of Sylow and Burnside, which will be proved in the module, and you will become familiar with different classes of groups, such as finite groups and dihedral groups. The module will give some of the flavour of the greatest achievement in group theory during the 20th century.
Objectives: By the end of the module students should be familiar with the topics listed above under `Contents'. 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.
Books: No specific books are recommended for this module. There are many groups on Group Theory in the library, and some of these might be helpful for parts of the module, but no single book is likely to cover the whole syllabus.