# Mathematics

## MTHM010 - Representation Theory of Finite Groups (2019)

MODULE TITLE CREDIT VALUE Representation Theory of Finite Groups 15 MTHM010 Dr Henri Johnston (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
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: MTH2002 Algebra or equivalent (i.e. ECM2711 Groups, Rings and Fields and ECM2712 Linear Algebra).

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
• Mascke’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
• Restriction
• Induction
• Examples of character tables
• Algebraic integers
• Burnside’s theorem
• Revision

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 33 117
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 Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Exam mark; results released online.
Coursework 1 10 10 hours All

Coursework mark; comments on script; outline solutions available.

Coursework 2 10 10 hours All

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
All above Written exam (100%) All Ref/Def period

RE-ASSESSMENT NOTES

Referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

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