## MATH3071 Groups and Symmetry

### 15 creditsClass Size: 108

Module manager: Dr Joao Faria Martins
Email: J.FariaMartins@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2022/23

### Pre-requisite qualifications

MATH2020 or MATH2022 or equivalent

### This module is mutually exclusive with

 MATH5071M Groups, Symmetry and Galois Theory

This module is not approved as a discovery module

### Module summary

Group theory is the mathematical theory of symmetry. Groups arise naturally in both pure and applied mathematics, for example in the study of permutations of sets, rotations and reflections of geometric objects, symmetries of physical systems and the description of molecules, crystals and materials.Groups have beautful applications to counting problems, in which objects are counted up to symmetry, answering questions like: "How many ways are there to colour the faces of a cube with m different colours, up to rotation of the cube?".

### Objectives

To review and develop the basic notions and theorems of group theory.
To introduce the notion of a group acting on a set and its properties.
To study the powerful Sylow theorems which give information on the structure of an arbitrary finite group in terms of the prime divisors of its order; to use group actions in this study.
To give an introduction to Pólya counting theory and how it is used to count objects up to symmetry.
To develop the skills of rigorous logical argument and problem-solving in the context of group theory and symmetry.

Learning outcomes
On completion of this module, students should be able to:
a) Prove and use basic results on groups, homomorphisms and quotients;
b) Prove and use basic results of group actions;
d) Represent a group by permutations;
e) Use Sylow's theorems to show that a group is not simple;
f) Apply Pólya counting theory to some simple counting problems.

### Syllabus

Revision of basic properties of groups.
Subgroups, homomorphisms, Lagrange's Theorem.
Symmetric group, sign of a permutation, cycle decomposition.
Isomorphism theorems.
Group actions, Cayley's Theorem, Orbit-Stabiliser Theorem, application to symmetric group Sylow theorems and applications, simple groups.
Burnside's Lemma, Pólya counting theory.

If time permits, extra topics to be chosen from: group presentations, symmetry groups of planar and three-dimensional figures.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

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