Module Catalogue 2014/15

MAS3202 : Foundations of Group Theory

  • Offered for Year: 2014/15
  • Module Leader(s): Prof. Sarah Rees
  • Owning School: Mathematics & Statistics
Semesters
Semester 1 Credit Value: 10
ECTS Credits: 5.0
Pre Requisites
Code Title
MAS2213Algebra
MAS2216Enumeration and Combinatorics
MAS2223Linear Algebra
Pre Requisite Comment

N/A

Co Requisites
Co Requisite Comment

None

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

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.

Learning Outcomes

Intended Knowledge Outcomes

Group axioms; examples of groups of symmetries, numbers, matrices, permutations, subgroups, homomorphisms, isomorphisms, group actions, Lagrange's, Cauchy's, Cayley's and Sylow's theorems.

Intended Skill Outcomes

Perform calculations with permutations; be familiar with symmetric, alternating, cyclic, and dihedral groups; check group axioms, verify subgroups, homomorphisms; calculate orbits and stabilizers of group actions; compute the number of orbits; calculate cosets; know and use Lagrange's, Cauchy's, Cayley's and Sylow's theorems.

Graduate Skills Framework

Graduate Skills Framework Applicable: Yes
  • Cognitive/Intellectual Skills
    • Numeracy : Assessed
  • Self Management
    • Personal Enterprise
      • Initiative : Present
      • Problem Solving : Assessed
  • Interaction
    • Communication
      • Written Other : Assessed

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
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 ActivitiesDrop-in/surgery61:006:00Drop-ins in the lecture room
Scheduled Learning And Teaching ActivitiesDrop-in/surgery240:000:00Office Hours in a staff office
Guided Independent StudyIndependent study121:3021:30Studying, practising and gaining understanding of course material
Guided Independent StudyIndependent study51:005:00Assignment review
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.

Reading Lists

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.

Timetable

Past Exam Papers

General Notes

N/A

Note: The Module Catalogue now reflects module information relating to academic year 14/15. Please contact your School Office if you require module information for a previous academic year.

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.