2005/06 Undergraduate Module Catalogue

MATH3071 Groups and Symmetry

15 creditsClass Size: 100

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

Pre-requisites

MATH2031 or equivalent.

This module is approved as an Elective

Objectives

To obtain some of the more important subgroup and structure theorems in the theory of groups. On completion of this module, students should be able to: a) calculate the full symmetry groups of simple geometric figures; b) represent such a group by permutations groups; c) Make use of direct products and Sylow?s Theorems in determining the structure of special groups. Including simple groups - and, possibly abelian and nilpotent groups; d) Prove (and be able to use) the homomorphism theorems and the Jordan-Hvlder theorem.

Syllabus

The theory of groups is the mathematical theory of symmetry - not only of geometric figures but also symmetries of sets - that is permutations. The ultimate aim of the theory is to describe (the structure) of all groups. This project can be completed in special cases, for example, if the group is finitely generated and abelian or if the group is finite and nilpotent. Where such description is not easy one may try to show that certain types of subgroup are present. Hence the importance of Sylow?s theorems (a kind of converse to Lagrange?s theorem). The idea of a group presentation arises naturally in many areas of mathematics - especially in algebraic topology. The topics covered are: Preliminaries. Symmetries of plane and solid figures; Permutation groups. Groups acting as sets; Factor groups; Homomorphism theorems. Direct products. Sylow?s theorems and applications plus topics from : Structure of finitely generated abelian groups; Structure of finite nilpotent groups (via Sylow subgroups). Free groups, generators and relations. Groups acting on graphs. Mathieu groups.

Teaching methods

Lectures: 26 hours. 7 examples classes.

Methods of assessment

One 3 hour examination at end of semester (100%).

Reading list

The reading list is available from the Library website

Last updated: 13/05/2005

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