### MAS3202 : Foundations of Group Theory

• Module Leader(s): Prof. Sarah Rees
• Owning School: Mathematics & Statistics
##### Semesters
 Semester 1 Credit Value: 10 ECTS Credits: 5.0

#### Aims

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.

#### Outline Of Syllabus

Symmetries. Definition of group. Groups of symmetries, groups of numbers, cyclic and dihedral groups, matrix groups. Subgroups, cosets, and Lagrange's Theorem. Symmetric and alternating groups. Isomorphisms, homomorphisms, quotient groups, and the Isomorphism Theorem. Group actions, Cauchy’s, Cayley's and Sylow’s theorems.

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion111:0011:00Revision for unseen Exam
Guided Independent StudyAssessment preparation and completion11:301:30Unseen Exam
Guided Independent StudyAssessment preparation and completion55:0025:00Written assignments
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Scheduled Learning And Teaching ActivitiesLecture61:006:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture221:0022:00Formal lectures
Scheduled Learning And Teaching ActivitiesDrop-in/surgery61:006:00Drop-ins in the lecture room
Guided Independent StudyIndependent study51:005:00Assignment review
Guided Independent StudyIndependent study121:3021:30Studying, practising and gaining understanding of course material
Total100:00
##### 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. Drop-ins are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. Office hours provide an opportunity for more direct contact between individual students and the lecturer.

#### Assessment Methods

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

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination901A90unseen
##### Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10N/A
##### Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. Written assignments (approximately 5 pieces of work of approximately equal weight) 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; this is thus formative as well as summative assessment.