Study information

Representation Theory of Finite Groups - 2024 entry

MODULE TITLERepresentation Theory of Finite Groups CREDIT VALUE15
MODULE CODEMTHM010 MODULE CONVENERDr Henri Johnston (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
Number of Students Taking Module (anticipated)
DESCRIPTION - summary of the module content

This course is an introduction to the representation theory of finite groups. We will develop the basic theory of finite dimensional representations over the complex numbers. A key result is that such representations are completely reducible and completely determined by their characters. We will also see how characters are used to effectively calculate such decompositions. We will study many examples.

Prerequisite modules: MTH2010 and MTH2011.

AIMS - intentions of the module

The aim of this module is to motivate and develop the basic theory of the representation theory of finite groups both as an abstract theory and through the study of important examples.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge:

1. State and apply key definitions in the representation theory of finite groups;

2. State, prove and apply core theorems in the representation theory of finite groups;

Discipline Specific Skills and Knowledge:

3. Perform computations accurately;

4. Use abstract reasoning to solve a range of problems;

Personal and Key Transferable / Employment Skills and Knowledge:

5. Think analytically and use logical argument and deduction;

6. Communicate your findings effectively in writing;
 
7. Work independently and manage your time and resources effectively.
SYLLABUS PLAN - summary of the structure and academic content of the module
  • Brief review of concepts from group and rings theory (groups, rings, homomorphisms, subgroups, subrings, normal subgroups)
  • Brief review of concepts from linear algebra (vector spaces, linear transformations)
  • Group representations
  • Group algebras
  • Modules over a group algebra
  • Maschke’s Theorem
  • Schur’s Lemma
  • Characters
  • Inner products of characters
  • The number of irreducible characters
  • Character tables
  • Orthogonality relations
  • Normal subgroups and lifted characters
  • Tensor products
  • Examples of character tables
  • Algebraic integers
  • Revision
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 33 Guided Independent Study 117 Placement / Study Abroad 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
Category Hours of study time Description
Scheduled learning and teaching activities 33 Lectures, including revision.
Guided independent study 117 Lecture and assessment preparation; wider reading; working on formative and summative questions.

 

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
       

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam 80 2 hours 1-7 Exam mark; results released online.
Coursework 1 10 10 hours 1-7

Coursework mark; comments on script; outline solutions available.
Coursework 2 10 10 hours 1-7

Coursework mark; comments on script; outline solutions available.

 

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Written exam Written exam (2 hours) 1-7 Referral/deferral period
Coursework 1 Coursework 1 1-7 Referral/deferral period
Coursework 2 Coursework 2 1-7 Referral/deferral period

 

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped. 

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 50%. 

 

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Gordon James and Martin Liebeck Representations and Characters of Groups Second edition CUP
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES MTH2010, MTH2011
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 7 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Wednesday 3rd April 2019 LAST REVISION DATE Monday 4th March 2024
KEY WORDS SEARCH Finite group, Field, Representation theory, Character, Character tables

Please note that all modules are subject to change, please get in touch if you have any questions about this module.