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Groups & Actions - MAT00075H

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• Department: Mathematics
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2022-23

Module summary

This is for postgraduate students only

Related modules

Additional information

MSc students should have taken a suitable first course in Pure Mathematics

Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23

Module aims

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 20^th Century mathematics, the Classification of Finite Simple Groups.

Module learning outcomes

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.

Module content

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.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

Pass/fail

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

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.

Indicative reading

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