Module Catalogue 2024/25

Pre Requisite Comment

Or MAS2707 Vector Spaces, Groups and Algorithms 2022/23

Co Requisite Comment

N/A

Aims

In this module students get to know group theory as a prototypical example of a mathematical theory. Motivated by the study of symmetry of physical or mathematical systems, one introduces the fundamental notion of a group. There is an abundance of examples. Then one investigates maps between groups which preserve structure (homomorphims), subgroups and quotient groups, as well as group actions. One aims to bring some order into the abundance of examples. This can be achieved via classification which is aided by structural theorems about groups (Lagrange’s, Cauchy’s, Cayley’s, Sylow’s theorems). In many of these theorem, the notion of a group action is fundamental.

This module builds on the elementary group theory seen in MAS2707. A guiding theme is the classification of groups of small order and of special classes of finite groups.

Outline Of Syllabus

We revise elementary concepts: subgroups, homomorphisms, isomorphisms, Lagrange’s Theorem. We meet new important classes of groups, such as cyclic groups and matrix groups.

We introduce normal subgroups and factor groups. We prove the Isomorphism Theorem which associates an isomorphism to each homomorphism. We classify finite abelian groups. We study group actions, and Cayley's theorem and apply group actions to prove Cauchy’s and Sylow’s theorems, which are partial converses to Lagrange’s. We discuss simple groups and extensions.

Intended Knowledge Outcomes

Consolidation of fundamental notions of group theory (group axioms, homomorphism, isomorphism, cosets) and of examples (symmetric groups, dihedral groups, matrix groups, groups of numbers, cyclic groups).

Normal subgroups, factor groups, isomorphism theorem.

Simple groups and extensions.

Classification of finite abelian groups.

Group actions: Orbit-Stabiliser Theorem, Class Equation.

Cauchy’s Theorem, Cayley’s Theorem, Sylow’s Theorems.

Classification of finite groups of small order.

Intended Skill Outcomes

Students will learn how to:

Calculate cosets;

Identify isomorphic groups in different realisations;

Classify groups of small order;

Calculate orbits and stabilisers of group actions;

Compute the number of orbits;

Apply Lagrange’s, Cauchy’s, Cayley’s and Sylow’s Theorems to obtain structural statements and classification results about groups.

Students will develop skills across the cognitive domain (Bloom’s taxonomy, 2001 revised edition): remember, understand, apply, analyse, evaluate and create.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

General Notes

N/A