Introductory description

The module is planned as a serious introduction into the subject of Theory of Groups. See outline syllabus.

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. However, results will be stated for infinite groups too whenever possible. In this course we will study group actions, Sylow's theorem and its various proofs, study direct and semidirect products of groups, use those to identify up to isomorphism various groups of relatively small orders, study the notion of soluble groups, state and prove Jordan-Holder Theorem.

Learning outcomes

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

• 1. Understand the notion of group actions.
• 2. Be able to state Sylow's Theorem and provide distinct proofs of this theorem.
• 3. Be able to apply Sylow's Theorems and its corollaries to show that A_n, n>4, is simple.
• 4. Use Sylow Theorems to demonstrate that certain finite groups are not simple.
• 5. Understand the notions of direct and semidirect products.
• 6. Be able to identify (up to isomorphisms) certain finite groups.
• 7. Undertand the notion of soluble groups.
• 8. State and prove Jordan-Holder Theorem.

Subject specific skills

This module lays a solid foundation to the study of group theory. It develops in depth the notions that allow one to
investigate the structure of finite and infinite groups.
Students taking the module will learn some of the techniques required for working on a large-scale research project. These techniques are partly theoretical, and partly computational.

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.