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  


Assessment components 


Running in 2023/24 (Please note that module timeslots are subject to change) 

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.
Prior learning requirements
None.
Available for Study Abroad? NO
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.
ToddCoxeter 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. CayleyHamilton Theorem.
(LO1  LO4)
Balance of independent study and scheduled teaching activity
The module will be taught by a mixture of lectures, workshops and selfstudy 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 selfawareness 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 tutorled) and will also provide additional materials to stimulate the student interest and broaden their horizons.
Learning outcomes
LO1: Understand the ToddCoxeter 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 LO13 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 LO15.
Bibliography
Core Text:
London Met Lecture Notes available on the WebLearn.
Recommended Reading:
Gorman, S.P. et al.; Lecture Notes in Groups. University of North London.
Jordan, C and Jordan, D(1994), Groups. E. Arnold.
Fraleigh, J.B. (1976); A first course in abstract algebra. AddisonWesley.
J. Hefferon. Linear Algebra [Online] [Accessed June 2014].
Available from: http://joshua.smcvt.edu/linearalgebra/book.pdf