## MATH3071 Groups and Symmetry

### 15 creditsClass Size: 100

Module manager: Dr A Parker
Email: parker@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisites

 MATH1022 Introductory Group Theory

This module is approved as an Elective

### Module summary

Informal description: 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. Applications of group theory range throughout mathematics.

### 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

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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 7 1.00 7.00 Lecture 26 1.00 26.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated