Groups and Symmetry

Module code: MA3131

Groups measure symmetry in the same way as numbers measure size. This is most evident in the study of symmetry in 2- and 3-dimensional geometric figures. Symmetry, and hence groups, play a key role in the study of crystallography, elementary particle physics, chemistry, coding theory, and the Rubik’s cube, to name just a few. This module develops the main ideas of symmetry and group theory, and the emphasis is on examples, structure theorems, classification results and decomposition concepts that have evolved as a result of attempts to describe all possible groups and symmetries. 

Although such a description is not actually feasible, it is possible to obtain surprisingly detailed information about the structure of large classes of groups. In particular, in this module we will discuss isometries of a Euclidean space, symmetry groups of regular polygons and platonic solids and finite rotation groups in 3d. We will also show how any group can be built from simple groups and develop the theory needed to present a rough idea of the statement of the classification of finite simple groups.


  • 33 hours of lectures
  • 2 hours of seminars
  • 11 hours of tutorials
  • 104 hours of guided independent study


  • Exam, 2 hours (70%)
  • Coursework (30%)