Please note that this module can be taken by some combined degree 3rd year students with the module code 0530011

Aims

• To introduce the notion of an axiomatic system through the example of group theory.
• To investigate elementary properties of groups.
• To illustrate these with a number of important examples, such as general linear groups and symmetric groups.

Learning objectives

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

• Know the axioms for a group.
• Recognise examples of groups.
• Know the definitions of basic terms, such as: order of a group, order of an element, subgroup, cyclic group and isomorphism.
• Prove simple consequences of the axioms, such as the cancellation law.
• Write elements of the symmetric group as cycles or products of transpositions, understand parity and be able to find symmetry groups of plane figures.
• Understand the proof, statement and simple uses of Lagrange's Theorem.
• Understand quotients and products of groups.
• Classify certain types of group.
• Understand the difference between finding a proof from the axioms that works for all groups, and finding a counterexample.

Syllabus

Definition of groups and examples.  Axioms, general linear group, symmetric groups, modular groups (revision from Core Algebra). 

Consequences of the group axioms. 

Subgroups.  Special linear group, alternating group, cyclic groups, symmetry groups of plane figures.

Homomorphisms.  The concept of a structure preserving map; homomorphisms, isomorphisms, automorphisms (also discussed in Linear Algebra for vector spaces). 

Lagrange.  Cosets and Lagrange's theorem. 

Generators.  Cyclic groups, dihedral groups (existence and maximality with respect to presentation). 

Fundamental theorem.  Normal subgroups, quotient groups and homomorphism theorems. 

Products.  External and internal direct products, classification of finite abelian groups (statement only). 

Classification of groups of small order. 

Rotations in 3 dimensions.  3D rotations and special orthogonal group. Finite subgroups. Platonic/regular solids and their symmetries. The ‘belt trick' and example. (if time allows)

Orthogonal and unitary groups.  (If time allows.  This is discussed more fully in Linear Algebra.)

Recommended texts

• C R and D A Jordan, Groups, Butterworth-Heinemann.
• M A Armstrong, Groups and Symmetry, Springer.
• P M Cohn, Classic Algebra, Wiley-Blackwell. 
• J E Humphreys, A Course in Group Theory, Oxford
• W Ledermann and A J Weir, Introduction to Group Theory, Longman

Teaching

• AutumnTerm
• 2 lectures per week
• Weekly Seminar.

Assessment

One and a half hour closed examination in week 1 SpringTerm.

Elective information

A second course on abstract algebra, assuming a basic knowledge of group theory. Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.

Prerequisites

• Foundations Taught in: Core Mathematics (09/10) or Core Algebra (10/11)