module specification

MA5031 - Group Theory and Vector Spaces (2023/24)

Module specification Module approved to run in 2023/24
Module title Group Theory and Vector Spaces
Module level Intermediate (05)
Credit rating for module 30
School School of Computing and Digital Media
Total study hours 300

 210 hours Guided independent study 90 hours Scheduled learning & teaching activities
Assessment components
Type Weighting Qualifying mark Description
Coursework 25%   Coursework
In-Course Test 25%   1-hour test
Unseen Examination 50%   Unseen Exam( 2 hours)
Running in 2023/24

(Please note that module timeslots are subject to change)
Period Campus Day Time Module Leader
Year North Monday Morning

Module summary

The topics covered in the first part of this module is to introduce students to the abstract algebraic structures of groups, which arise from the ideas of symmetries and of vector and matrix calculus respectively. These two primary examples of algebraic structures have applications across science and engineering, and also provide a firm foundation of necessary basic algebraic notions for the student to further their study mathematical study.

The aim of the second part of the module is to introduce students to the abstract algebraic structures of vector spaces, developing on the material on linear algebra learnt previously. This primary example of algebraic structures has applications across science and engineering, and also provides a firm foundation of necessary basic algebraic notions for the student to further their mathematical study.

None.

Syllabus

Revision of notions of binary operation, associativity.

Formal definition of a group. Worked examples: symmetries of a regular polygon.

Generators, orders. Abelian groups, cyclic groups, free groups, relations in groups.

Subgroups. Cosets and Theorem of Lagrange. Normal subgroups; quotient groups.

Todd-Coxeter method for calculating order of a group and finding coset transversals.

Homomorphisms and isomorphisms. Kernel and image; first isomorphism theorem.

Permuation groups: symmetric and alternating groups. Cayley's theorem. Groups of matrices.

Vector Spaces: Revision of matrices and determinants. Formal definition and further examples of vector spaces. Linear combination, spanning, independence. Subspaces. Bases and dimensions, change of basis. Quotient spaces. Linear transformations, matrix representation.  Kernel and image; first isomorphism theorem, rank/nullity. Eigenvalues and bases of eigenvectors, diagonalisation. Cayley-Hamilton Theorem.

(LO1 - LO4)

Balance of independent study and scheduled teaching activity

The module will be taught by a mixture of lectures, workshops and self-study practical exercises. The lectures will be used to introduce the various concepts and principles of the module and their strengths in applications. Lectures will be followed by workshops.

The workshops will afford students the opportunity to work in small groups on exercises related to previously taught material. The students will be able to present previously completed exercises for comment from the lecturer and other students.  In this class time students will also be encouraged to explore and experiment with the concepts and techniques to encourage their own sense of mathematical creativity.

Students will be expected to spend time on unsupervised work, for example, private study of problem sheets and in the preparation of coursework (219 hours).  A framework will be put in place to encourage disciplined learning through student self-awareness of progress in volume of work, understanding, attendance and punctuality.

In addition to standard VLE presence there will be links available for further readings and discussion groups. Blended learning is incorporated by using online resources as a medium for communication (both peer and tutor-led) and will also provide additional materials to stimulate the student interest and broaden their horizons.

Learning outcomes

LO1: Understand the Todd-Coxeter method and apply it to gain further insights into the structures of groups.

LO2: Appreciate notions and theory of subgroups and quotient groups and apply them correctly to well defined problems and construct homomorphisms between groups and determine their properties.

LO3: Perform calculations with vectors and transformations using suitable matrix representations. Construct homomorphisms vector spaces and determine their properties such as rank and nullity.

LO4: Understand both the concrete examples and the concept of linear independence and be able to apply them to find bases from given spanning sets. Determine
whether a transformation of a vector spaces may be represented by a diagonal matrix with respect to an appropriate basis.

Assessment strategy

There will be one progression test covering LO1-3 which will give students opportunity to demonstrate their understanding of selection of topics. The Coursework will assess L04. The final assessment will be an exam where students will be tested on LO1-5.

Bibliography

Core Text:

London Met Lecture Notes available on the WebLearn.