# MAS8701 : Foundations of group theory

• Offered for Year: 2024/25
• Module Leader(s): Dr Stefan Kolb
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
##### Semesters

Your programme is made up of credits, the total differs on programme to programme.

 Semester 1 Credit Value: 10 ECTS Credits: 5.0 European Credit Transfer System

#### 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.

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture201:0020:00Formal lectures
Guided Independent StudyAssessment preparation and completion151:0015:00Completion of in course assessments
Scheduled Learning And Teaching ActivitiesLecture51:005:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Guided Independent StudyIndependent study581:0058:00Preparation time for lectures, background reading, coursework review
Total100:00
##### Jointly Taught With
Code Title
MAS3701Foundations of group theory
##### 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 Methods

The format of resits will be determined by the Board of Examiners

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination1201A80N/A
##### Exam Pairings
Module Code Module Title Semester Comment
Foundations of group theory1N/A
##### Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M5Problem-solving exercises assessment
Prob solv exercises1M5Problem-solving exercises assessment
Prob solv exercises1M5Problem-solving exercises assessment
Prob solv exercises1M5Problem-solving exercises assessment
##### 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 the learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

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.