Group Theory

Overview

- definition and examples of groups and their properties
- countability of a group and index
- Lagrange’s theorem
- normal subgroups and quotient groups
- group homomorphisms and isomorphism theorems
- structure of finite abelian groups
- Cayley’s theorem
- Sylow’s theorem
- composition series and solvable groups

Learning Objectives

It is intended that students shall, on successful completion of the module, be able to: understand the ideas of binary operation, associativity, commutativity, identity and inverse; reproduce the axioms for a group and basic results derived from these; understand the groups arising from various operations including modular addition or multiplication of integers, matrix multiplication, function composition and symmetries of geometric objects; understand the concept of isomorphic groups and establish isomorphism, or otherwise, of specific groups; understand the concepts of conjugacy and commutators; understand the subgroup criteria and determine whether they are satisfied in specific cases; understand the concepts of cosets and index; prove Lagrange's theorem and related results; understand the concepts and basic properties of normal subgroups, internal products, direct and semi-direct products, and factor groups; establish and apply the fundamental results about homomorphisms - including the first, second and third isomorphism theorems - and test specific functions for the homomorphism property; perform various computations on permutations, including decomposition into disjoint cycles and evaluation of order; apply Sylow's theorem.

Skills

Numeracy and analytic argument skills, problem solving, analysis and construction of proofs.

Assessment

None