Aims

The aim of the module is to develop the theory of finite groups up to the Sylow Theorems and to study factorisation in commutative integral domains.

Learning objectives

At the end of the module you should be able to...

• Work with group actions;
• Use Sylow's Theorems;
• Recognise examples of various types of integral domain;
• Use Euclid's algorithm in Euclidean domains.

Syllabus

• Revision of basic group theory, finite abelian groups.  Group actions: orbits, stabilisers. Conjugation and the class equation; conjugacy in Sn and the simplicity of An. p-groups. Orbit counting theorem and examples. Finite direct products. Sylow Theorems with proofs and applications. Non-simplicity tests and techniques for showing that certain numbers cannot be the orders of simple groups.
• Revision of basic facts from ring theory: homomorphisms, maximal and prime ideals, integral domains and fields. Irreducible and prime elements. Different types of integral domain: Unique Factorization Domains, Principal Ideal Domains, Euclidean Domains. Every Euclidean Domain is a Principal Ideal Domain. Every Principal Ideal Domain is a Unique Factorization Domain. Illustrative examples including the integers, Gaussian integers and polynomials over a field.

Recommended texts

• J B Fraleigh, A First Course in Abstract Algebra, Addison-Wesley

Teaching

• Spring: Weeks 2-10 Three hours per week
• Summer: Weeks 2-5 Four hours per week

Roughly two thirds of the contact time in this course is devoted to lectures. The other third consists of problem sessions, examples classes, revision, etc.

Assessment

Three hour unseen examination towards the end of the summer term 90%

Coursework 10%. Note that coursework submitted after the advertised deadlines will be given a mark of zero.

Elective information

This module is not available as an elective.

Prerequisites

• all Taught in: Groups and Rings