MAS3202 : Foundations of Group Theory

  • Module Leader(s): Dr Mark Grant
  • 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. The main emphasis is on finite groups. We introduce the concept of isomorphism of groups and classify groups of small order up to isomorphism, introducing the necessary tools as we go along: Basic properties of inverse elements, subgroups, powers of elements, element orders, cyclic groups, cosets, Lagrange's Theorem, Cartesian products, conjugacy classes, the class equation, homomorphisms, and the Isomorphism Theorem.

There will be many examples of groups of symmetries, numbers, matrices, and permutations. In particular, symmetric and alternating groups will be introduced and studied. We will show cycle decomposition and sign of permutations.

We will also introduce group actions and show how they can be used for counting arguments.

Outline Of Syllabus

Group axioms, isomorphisms, groups of small order, inverse elements, subgroups, powers of elements, element orders, cyclic groups, cosets, Lagrange's Theorem, Cartesian products, conjugacy classes, the class equation, homomorphisms, the Isomorphism Theorem, groups of symmetries, numbers, matrices, and permutations, symmetric and alternating groups, cycle decomposition and sign of permutations, group actions, counting arguments.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture61:006:00Problem classes
Scheduled Learning And Teaching ActivitiesLecture221:0022:00Formal lectures
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 ActivitiesDrop-in/surgery240:000:00Office Hours in a staff office
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.

Reading Lists

Timetable

Disclaimer: The University will use all reasonable endeavours to deliver modules in accordance with the descriptions set out in this catalogue. Every effort has been made to ensure the accuracy of the information, however, the University reserves the right to introduce changes to the information given including the addition, withdrawal or restructuring of modules if it considers such action to be necessary.