# Groups, Rings and Modules

## Description

#### Overview

Within the Structure & Symmetry theme, there are two independent algebra modules that together comprise a basic introduction to abstract algebra.  You can take either one, neither, or both.

In Groups, Rings & Modules (this module, first semester) we study groups and commutative rings which are maybe the most natural algebraic structures. We assume you are familiar with the most simple concepts of group theory as they are taught in compulsory algebra courses everywhere, and cover a standard set of topics in the structure theory especially of finite groups.

In the commutative rings part we discuss both of the two classical motivations for studying commutative rings: number theory on the one hand and algebraic geometry on the other.

In Algebras & Representation Theory (second semester) we turn to noncommutative rings. Here an important motivation are matrix rings and linear algebra. In fact the first lectures will be still about commutative rings, but with an application in linear algebra (the Jordan normal form). The highlight is then one of the main theorems in noncommutative ring theory, the Artin-Wedderbuurn theorem, which can be seen as an abstraction of some of the ideas developed in commutative algebra to noncommutative rings.

Finally we return to groups and study their representations which is a subject of immense importance both in pure mathematics and in physcis or chemistry. Again, the material covered is fairly standard and culminates in one of the applications of representation theory in abstract group theory, which is Burnside's theorem.

#### Groups (5 lectures)

• Simple groups, Jordan-Hölder Theorem, (semi)direct products
• Permutation representations and group actions
• Sylow Thorems and applications
• Abelian, soluble and nilpotent groups
• Free groups and presentations

Commutative rings (5 lectures)

• Modules: introduction
• Chain conditions and Hilbert's basis theorem
• Fields and numbers
• Affine algebraic geometry and Hilbert's Nullstellensatz
##### Assessment

This module is assessed in two assignments.

##### Prerequisites

You should be familiar and comfortable with:
Basic linear algebra; Definitions and examples of groups, rings, fields; Basic algebra concepts such as homomorphisms; Basic notions of group theory - permutations, symmetric groups, Lagrange's theorem, normal subgroups and factor groups