Synopsis for Group Theory

Overview

This group theory course develops the theory begun in mods. In this course we will use groups to analyse symmetries. Groups first appeared in the study of symmetries of roots of polynomials and now have many applications in physics and other sciences as symmetry groups. The main concept that will be introduced in the course is that of a group acting on a set.

Learning Outcomes

Students will gain a deeper understanding of the notion of symmetry. They will learn to use actions of groups and the counting formula. As an application they will see the classification of finite subgroups of the rotation group which is equivalent to classifying platonic solids.

Synopsis

The group of motions of the euclidean plane (1 lecture). Finite groups of motions (1 lecture).

Group actions (1 lecture). Actions on cosets (1 lecture). Permutation representations and Cayley's theorem (1 lecture).

The orbit-counting formula (1 lecture). Finite subgroups of SO(3). Platonic solids and their symmetries (sketch of proof) (1 lecture). The conjugation action and the class equation (1 lecture).

1. Michael Artin, Algebra (1st ed. Pearson, 1991) Chapters 5,6 P.J.
2. Cameron Introduction to Algebra (2nd. ed., OUP, 2008) pp. 124-146, 237-250.

   Further Reading

3. Peter M. Neumann, G. A. Stoy, E. C. Thompson, Groups and Geometry (OUP, 1994, reprinted 2002), ISBN 0-19-853451-5. Chapters 1-9, 15.
4. Geoff Smith, Olga Tabachnikova, Topics in Groups Theory (Springer Undergraduate Mathematics Series, 2002) ISBN 1-85233-2. Chapter 3.
5. M. A. Armstrong, Groups and Symmetry (Springer, 1988), ISBN 0-387-96675-7, Chapters 1-19.
6. Joseph J. Rotman, A First Course in Algebra (Second Edition, Prentice Hall, 2000), Chapter 2.