Department of Mathematics

MA3131 Group Theory

Subject Knowledge

Aims

Learning Outcomes

• the concepts of homomorphisms, normal subgroups and quotient groups and their relevance to the structure of a group;
• the idea of a group action and how group actions are used for enumeration and to prove fundamental results such as the Sylow theorems;
• the idea of a group presentation and how to calculate using generators and relations;
• the properties of permutations and of symmetric and alternating groups;
• the idea of simple groups as the basic building blocks of group theory.
• composition series and chief series for groups, the Jordan-Hölder Theorem and solvable and nilpotent groups.

Methods

Assessment

The exam paper contains 8 questions. Any number of questions may be attempted, but only the best 5 answers will be taken into account. Full marks may be obtained for answers to 5 questions. All questions carry equal weight.

Subject Skills

Aims

Learning Outcomes

Methods

Assessment

Explanation of Pre-requisites

Course Description

One of the major goals of the module is to develop enough theory to be able to discuss the classification of the finite simple groups, at least in broad general terms. This classification ranks as one of the major achievements in pure mathematics. Its proof runs to somewhere between 10,000 and 15,000 journal pages, spread across some 500 separate articles by more than 100 mathematicians, almost all written between 1950 and the early 1980's. A revision is currently underway, but even this is expected to run to more than 5,000 pages.

The module is designed to present a broad outline of the classification, to explain its significance, and to give a hint of the complexity of its proof.

Syllabus

Definition and examples of groups; the symmetric group and cycle decomposition of elements; free groups; subgroups; centralisers; centre of a group; products of subsets; cosets; Lagrange's Theorem; conjugates and conjugacy classes of elements and of subsets; normal subgroups; normalisers; quotient groups; homomorphisms and isomorphisms; kernels and images; the isomorphism theorems; normal closures; presentations of groups; generators and relations; direct products; automorphism groups and the conjugation map;

-sets and actions; the kernel of an action; faithful and transitive actions; multiplication and conjugation actions; stabilisers and orbits; the Orbit-Stabiliser Theorem; the fixed point theorem for the number of orbits of an action; use of this theorem for enumeration. The alternating groups ; Cayley's theorem that every group is isomorphic to a subgroup of a symmetric group;

Sylow subgroups of finite groups; the Sylow theorems; characteristic subgroups; simple groups; normal and subnormal series; composition series and chief series; the Jordan-Hölder Theorem; solvable and nilpotent groups; commutator subgroups; upper and lower central series; derived series; that a finite -group is nilpotent; normalisers in nilpotent groups; characterisation of nilpotent groups; simplicity of the alternating groups for ; outline of the classification of finite simple groups.

Reading list

Recommended:

J. F. Humphreys, A Course in Group Theory, Oxford University Press.

Background:

J. A. Gallian, Contemporary Abstract Algebra, DC Heath.

D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, 40.1. American Mathematical Society.

D. L. Johnson, Presentations of Groups, Cambridge University Press.

C. R. Jordan and D. A. Jordan, Groups, Edward Arnold.

I. D. MacDonald, Theory of Groups, Oxford University Press.

J. S. Rose, A Course in Group Theory, Cambridge University Press.

J. J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag.

Resources

Module Evaluation

Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.