This module is not currently running in 2024 to 2025.
Groups arise naturally in many areas of mathematics as well as in chemistry and physics. A concrete way to approach groups is by representing them as a group of matrices, in which explicit computations are easy. This approach has been very fruitful in developing our understanding of groups over the last century. It also helps students to understand aspects of their mathematical education in a broader context, in particular concepts from earlier modules (From Geometry to Algebra/Groups and Symmetries and Linear Algebra) have been amalgamated into more general and powerful tools.
This module will provide a rigorous introduction to the main ideas and notions of groups and representations. It will also have a strong computational strand: a large part of the module will be devoted to explicit computations of representations and character tables (a table of complex numbers associated to any finite group).
At level 7, topics will be studied and assessed to greater depth.
Total contact hours: 42
Private study hours: 108
Total study hours: 150
Level 6
Assessment 1 (10-15 hrs) 20%
Assessment 2 (10-15 hrs) 20%
Examination (2 hours) 60%
Level 7
Assessment 1 (10-15 hrs) 20%
Assessment 2 (10-15 hrs) 20%
Examination (2 hours) 60%
Reassessment methods:
Like-for-like
We will not follow a single text, and the lecture notes will cover the entire syllabus. Nevertheless
G.D. James and M. Liebeck, Representations and characters of groups, CUP (2001)
J.P. Serre, Linear representations of finite groups, Springer GTM (1977)
J.L. Alperin and R.B. Bell, Groups and Representations, Springer GTM (1995)
contain a large amount of the material.
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes. On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of the theory and practice of groups (with examples including permutation groups and matrix groups, and the combinatorics of the
symmetric group), of linear algebra, and of representations and characters of groups.
2 demonstrate the capability to solve complex problems using a very good level of skill in calculation and manipulation of the material in the following areas: calculations
within permutation groups and matrix groups; computations of the character tables of small groups; derivation of structural information about a group from its character
table;
formulation and proof of simple statements about groups and representations in precise abstract algebraic language; breaking up representations into smaller simpler
objects; composition series and composition factors of small groups.
3 apply a range of concepts and principles in group theory and representation theory in loosely defined contexts, showing good judgment in the selection and application of
tools and techniques.
The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed
2 demonstrate a high level of capability in developing and evaluating logical arguments
3 communicate arguments confidently with the effective and accurate conveyance of conclusions
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working
5 solve problems relating to qualitative and quantitative information
6 make effective use of information technology skills such as online resources (moodle) and internet communication.
7 communicate technical material effectively
8 demonstrate an increased level of skill in numeracy and computation.
9 demonstrate the acquisition of the study skills needed for continuing professional development
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