Module manager: Kevin Houston
Email: K.Houston@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running 2026/27
None
| MATH1120 | Introduction to Group Theory |
| MATH3071 | Groups and Symmetry |
MATH3071 Groups and Symmetry
This module is not approved as a discovery module
Group theory is the mathematical study of symmetry. Groups arisenaturally in both pure and applied mathematics, for example in the studyof permutations of sets, rotations and reflections of geometric objects,symmetries of physical systems and the description of molecules, crystalsand materials. They also have beautiful applications in countingproblems, where we count objects up to symmetry. For example, grouptheory can answer questions like: "How many distinct ways can the facesof a cube be coloured with m different colours, up to rotation of thecube?"
On completion of this module, students should be able to ...
To review and develop the basic notions and theorems of group theory.
To introduce the notion of homomorphism.
To introduce the notion of quotient group.
To recall the notion of a group acting on a set and to develop its properties and applications.
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 introduce 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.
On successful completion of the module students will be able to:
1. Prove and use basic results on groups, subgroups, and homomorphisms.
2. Represent a group by permutations;
3. Calculate quotient groups and prove theorems about them;
4. Prove and use basic results of group actions;
5. State and use the Orbit-Stabiliser Theorem;
6. Apply Pólya counting theory and Burnside’s Lemma to somesimple counting problems;7. Use Sylow's theorems to show that a group is not simple.
On successful completion of the module students will be able to:
a) Creatively solve unseen problems
b) Communicate technical mathematics in written form
c) Identify areas required for improvement
d) Develop critical thinking
e) Manage workload
Revision of basic properties of groups.
Subgroups, Lagrange's Theorem.
Symmetric group, sign of a permutation, cycle decomposition
.Homomorphisms.
Quotient groups and isomorphism theorems.
Group actions.
Orbit-Stabiliser Theorem, application to symmetric group, Cayley'sTheorem.
Conjugacy, centralisers.
Burnside's Lemma, Pólya counting theory.
Sylow theorems and applications, simple groups.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lectures | 44 | 1 | 44 |
| Private study hours | 156 | ||
| Total Contact hours | 44 | ||
| Total hours (100hr per 10 credits) | 200 | ||
Regular pieces of coursework which students may submit for formative feedback
Check the module area in Minerva for your reading list
Last updated: 30/04/2026
Errors, omissions, failed links etc should be notified to the Catalogue Team