# MAS8701 : Foundations of group theory

• Offered for Year: 2021/22
• Module Leader(s): Professor Sarah Rees
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
##### Semesters
 Semester 1 Credit Value: 10 ECTS Credits: 5

#### Aims

To introduce students to the basic ideas of group theory.

Module Summary

The module introduces the axioms of a group and explores many examples, particularly of symmetry groups,permutation groups, and matrix groups. Basic properties of groups will be demonstrated, always with reference to examples.

We shall prove Lagrange's Theorem, which tells us that for finite groups the number of elements in a subgroup divides the number of elements in the parent group. Groups of permutations will be studied systematically.
We will introduce group homomorphisms and prove the Isomorphism Theorem which associates an
isomorphism to each homomorphism. We also consider 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.

#### Outline Of Syllabus

To introduce students to the basic ideas of group theory.

Module Summary

The course introduces the axioms of a group and explores many examples, particularly of symmetry groups,permutation groups, and matrix groups. Basic properties of groups will be demonstrated, always with reference to examples.
We shall prove Lagrange's Theorem, which tells us that for finite groups the number of elements in a subgroup divides the number of elements in the parent group. Groups of permutations will be studied systematically.
We will introduce group homomorphisms and prove the Isomorphism Theorem which associates an
isomorphism to each homomorphism. We also consider 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.

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture201:0020:00Formal lectures - Present in Person
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures - Present in Person
Scheduled Learning And Teaching ActivitiesLecture51:005:00Problem classes - Synchronous On-line
Guided Independent StudyAssessment preparation and completion151:0015:00Completion of in course assessments
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

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
MAS3701Foundations of group theory1N/A
##### Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10Coursework assignments
Prob solv exercises1M10Course work assignments
##### Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. 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.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.