- Department: Mathematics
- Credit value: 10 credits
- Credit level: H
- Academic year of delivery: 2022-23
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
Pre-requisite modules: students must have taken Pure Mathematics or Pure Mathematics Option 1.
Occurrence | Teaching period |
---|---|
A | Autumn Term 2022-23 |
This module is designed as an “exit level” module for H-level undergraduate students and also as an introduction to ideas which will be very important to many of the students who go on to further study in many areas of mathematics. As the study of symmetry, Group Theory is ubiquitous in mathematics, and has applications across science more generally. This module will introduce students to the idea of group actions, and then move on to the important Sylow Theorems, ending with a taste of one of the great achievements of 20th Century mathematics, the Classification of Finite Simple Groups.
Subject content
Group actions
The Sylow Theorems
Conjugacy in groups
Simple groups
Academic and graduate skills
The material in this module is central to any undergraduate programme involving algebra, and will also be of use to students in subsequent modules (for example, Galois Theory).
For students going on to further study exposure to this material is an essential part of a good mathematical training.
For all students, by the end of the module they will be able to understand one of the crowning achievements of pure mathematics in the last 100 years.
Outline syllabus:
Recap of first isomorphism theorem and derivation of the others
Definition of a group action. The Orbit-Stabilizer Theorem.
Burnside’s Orbit Counting Lemma
The Sylow Theorems
Conjugacy Classes. Illustration in symmetric and alternating groups.
The Jordan-Holder theorem
Simple groups. Simplicity tests using Sylow theorems and actions, overview of the Classification of Finite Simple Groups, and examples. A_n is simple for n > 4.
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
None
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
Groups And Symmetry, M. Armstrong, Springer Undergraduate texts
A course in Group Theory, J.F. Humphreys, OUP
An introduction to the theory of Groups, J.J. Rotman, Springer Graduate Texts