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Group Theory (MATH 33300)

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Administrative Information

  1. Unit number and title: MATH 33300 Group Theory
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: before 1990
  6. Unit Organiser: Jeremy Rickard
  7. Lecturer: Prof Jeremy Rickard
  8. Teaching block: 1
  9. Prerequisites: MATH11511 Number Theory & Group Theory and MATH11006 Analysis 1

Unit 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

General Description of the Unit

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.

Teaching Methods

Lectures and exercises to be done by the students.

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.

Assessment Methods

The assessment mark for Group Theory is calculated from a 2½-hour written examination in January 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.

Award of Credit Points

To gain credit points, you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

Transferable Skills

Assimilation and use of novel and abstract ideas.

Texts

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.

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

Unit Webpage

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