2017/18 Undergraduate Module Catalogue

MATH3071 Groups and Symmetry

15 creditsClass Size: 50

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

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2017/18

Pre-requisite qualifications

MATH2020 or MATH2022 or equivalent

This module is approved as a discovery module

Module summary

Informal description: The theory of groups is the mathematical theory of symmetry. Here symmetry is interpreted in a very broad sense, e.g. symmetries of sets (permutations) or symmetries of planar and three-dimensional figures. Group theory and symmetry are intrinsically important in their own right. They are also important because of their ubiquity throughout all areas of mathematics, for instance geometry, topology, differential equations, representation theory and combinatorics. Group theory also has applicationswell beyond pure and applied mathematics, frequently appearing in Physics (for example dealing with symmetries not only of physical systems but also of physical theories themselves) and Chemistry, including describing the structure of molecules, crystals and materials. The first part of the course will be a review of some group theory notions from MATH2022 Groups and Vector Spaces (e.g. groups, subgroups, quotient groups and symmetric groups). We will then introduce the very useful concept of symmetry in the context of group actions, which will streamline the rest of the course. We will then move on to study some impressive general results surrounding the structure of finite groups, namely Sylow theory. This course will culminate with Pólya counting theory, in which group symmetry theoretical techniques are used to count objects up to symmetry: e.g. “How many ways are there to colour the faces of a cube with m different colours, up to rotation of the cube?”.

Objectives

On completion of this module, students should be able to:
a) prove (and be able to use) some basic results on groups, homomorphisms and group quotients;
b) prove (and be able to use) the group isomorphism theorems;
c) prove (and be able to use) the basic results of group actions, including the Orbit-stabiliser Theorem,
d) represent a group by permutations;
e) make use of direct / semi-direct products and Sylow's theorems in determining the structure of special groups, including determining when groups cannot be simple;
f) apply Burnside's Lemma and general Pólya counting theory to some simple counting problems.

Syllabus

- Preliminaries: groups, subgroups, homomorphisms, quotient groups, Lagrange's theorem.
- Review of the symmetric group: sign of a permutation, cycle decomposition.
- First, second and third isomorphism theorems.
- Groups acting on sets: Cayley's theorem; orbits and stabiliser subgroups; orbit-stabiliser theorem; homomorphisms and isomorphisms of G-sets. Applications to the symmetric group; cycle structure.
- Sylow theory: Cauchy's theorem, the conjugacy class equations, Sylow's theorems, applications of Sylow's structure theory for groups. Simple groups.
- Burnside's Lemma and Pólya counting theory.
- If time permits, extra topics may be chosen from the following list: Composition series of finite groups, Jordan-Hölder Theorem, soluble groups. Group presentations. Abelianisations. 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

Reading list

The reading list is available from the Library website

Last updated: 26/04/2017

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