# Group Theory

## Unit aims

To develop the student's understanding of groups, one of mathematics' most fundamental constructs.

## Unit description

Groups are one of the main building blocks in mathematics. They form the basis of all rings, fields and vector spaces, and many objects studied in analysis and topology have a group-theoretic structure. Also, physicists use groups to describe properties of the fundamental particles of matter. Pure mathematicians use them to study symmetry properties of geometric figures, in problems concerning permutations, to classify sets of objects like points of algebraic curves, and to study collections of matrices as well as in many other uses. The unit will cover the basic parts of the subject and study finite groups in some detail.

## Relation to other units

This unit develops the Group Theory material in Level C/4 Pure Mathematics. The ideas are carried further in the Level M/7 units Representation Theory, Algebraic Topology, and Galois Theory.

## Learning objectives

After taking this unit, students should have gained an understanding of the basic properties of finite groups and an appreciation of the beauties of the subject and the limits of our present understanding.

## Transferable skills

Assimilation and use of novel and abstract ideas.

## Syllabus

1. Basics concepts
2. Subgroups
3. Cosets and quotient groups
4. Symmetric groups
5. Products
6. Group actions
7. Sylow’s Theorems
8. Applications of Sylow’s Theorems
9. Soluble groups
10. The Jordan-Holder Theorem
11. The Classification Theorem

## Reading and References

A Course in Group Theory (OUP) by John F. Humphreys.

A Course on Finite Groups (Springer) by Harvey E. Rose

Printed notes will be provided.

Unit code: MATH33300
Level of study: H/6
Credit points: 20
Teaching block (weeks): 1 (1-12)
Lecturer: Dr Tim Burness

## Pre-requisites

MATH10005 Introduction to Group Theory and MATH10003 Analysis 1A. One of MATH 21800 Algebra 2 or MATH 21100 Linear Algebra 2 is desirable, but not essential.

None

## Methods of teaching

Lectures and exercises to be done by the students.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

• 90% from a 2 hour 30 minute exam
• 10% from assigned homework questions

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.