# Group Theory

## Overview

• Credit value: 15 credits at Level 7
• Assessment: a three-hour examination (80%) and short, problem-based assignments (20%)

## Module description

This module introduce you to further topics in group theory, extending and building on material covered at undergraduate level. It is a stand-alone course but will also give you experience in the subject at postgraduate level which will be useful if you are intending to pursue research in this area.

### indicative module Syllabus

• Group actions and G-sets; the Orbit-Stabilizer Theorem; application of G-sets to the proof of Sylow’s Theorems; use of Sylow’s Theorems to investigate groups of finite order; discussion of simple groups
• Direct products of groups; groups which are the direct product of their Sylow subgroups; classification of finite abelian groups as direct products of certain cyclic groups
• Revision of quotient groups; the three Isomorphism Theorems; normal series, composition series, chief series; the Jordan-Holder Theorem; composition factors and chief factors; definition of soluble groups; showing that a given group is or is not soluble; examples

## Learning objectives

By the end of this module, you will have:

• knowledge and understanding of, and the ability to use, mathematical techniques
• knowledge and understanding of a range of results in mathematics
• a deeper understanding of several topics in mathematics
• the ability to comprehend and evaluate conceptual and abstract material
• the ability to devise rigorous mathematical proofs
• the ability to understand and apply mathematical reasoning in several different areas of mathematics
• problem-solving skills, including the ability to assess problems logically and to approach them analytically.