Unit information: Group Theory in 2012/13

Unit name	Group Theory
Unit code	MATH33300
Credit points	20
Level of study	H/6
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Professor. Rickard
Open unit status	Not open
Pre-requisites	Level 1 Pure Mathematics
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

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.

Aims

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

Unit Webpage: http://www.maths.bris.ac.uk/~majcr/groupthy.html

Syllabus

1. Basics Concepts
2. Homomorphisms
3. Subgroups
4. Generators
5. Cyclic groups
6. Cosets and Lagrange’s Theorem
7. Normal subgroups and quotient groups
8. Isomorphism Theorems
9. Direct products
10. Group actions
11. Sylow’s Theorems
12. Applications of Sylow’s Theorems
13. Finitely generated abelian groups
14. The symmetric group
15. The Jordan-Holder Theorem
16. Soluble groups
17. Free groups

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.

Intended Learning Outcomes

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.

Teaching Information

Lectures and exercises to be done by the students.

Assessment Information

The assessment mark for Group Theory is calculated from a 2½-hour written examination in April consisting consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted be used in this examination.

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.