Content: 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.

This module will focus on laying the foundation for the study of modern group theory. The notions of group actions fundamental to the subject will be investigated in depth.
You will become familiar with different classes of groups such as finite groups, dihedral groups, simple groups,
soluble groups. Techniques will include the theorems of Sylow and Jordan-Holder, which will be proved in the module.
Distinct proofs of these results will demonstrate different technical approaches. The module will give some of the flavour of the modern group theory.

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

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