Synopsis for Group Theory and an Introduction to Character Theory

Recommended Prerequisites: Part A Group Theory is essential. Part B Introduction to Representation Theory is useful as “further algebraic thinking”, and some of the results proved in that course will be stated (but not proved). In particular, character theory will be developed using Wedderburn's theorem (with Maschke's theorem assumed in the background). Students should also be well acquainted with linear algebra, especially inner products and conditions for the diagonalisability of matrices.

Overview

A finite group represents one of the simplest algebraic objects, having just one operation on a finite set, and historically groups arose from the study of permutations or, more generally, sets of bijective functions on a set closed under composition. Thus there is the scope for both a rich theory and a wide source of examples. Some of this has been seen in the Part A course Group Theory, and this course will build on that. In particular, the Jordan–H{ö}lder theorem (covered there but not examined) shows that there are essentially two problems, to find the finite simple groups, and to learn how to put them together. The first of these dominated the second half of the 20th century and has been completed; this proved a massive task, encompassing in excess of 20,000 printed pages, and much remains to be done to distil the underlying ideas. In this course, our aim will be to introduce some of the very fundamental ideas that made this work possible. Much is classical, but it will be presented in a modern form.

Learning Outcomes

By the end of this course, a student should feel comfortable with a number of techniques for studying finite groups, appreciate certain classes of finite simple groups that represent prototypes for almost all finite simple groups, and have seen the proofs of some of the "great" theorems. MFoCS students may be expected to read beyond the confines of the lectures from the “more sophisticated books" below to attempt the additional problems set for MFoCS students in preparation for working on the miniproject.

Synopsis

Review of isomorphism theorems (up to Jordan–H{ö}lder), composition series, soluble groups; some examples of groups of (relatively) small order. Constructions of groups; semidirect products, notion of presentations. Cauchy's theorem, Sylow's theorems, the Frattini argument. Orthogonality relations, construction of character tables and applications. The character ring as a subring of the algebra of complex-valued class functions. Burnside's -theorem. Some simple groups.

The bulk of the course is well covered by the web notes written in 2011 by Jan Grabowski, which will be reposted with minor corrections. (A few topics from those notes will not be covered as will be clear from this year's synopsis.)

Additional Reading

1. Geoff Smith and Olga Tabachnikova, Topics in Group Theory, Springer Undergraduate Mathematics Series (Springer-Verlag, 2000). ISBN 1-85233-235-2
2. H. Kurzweil and B. Stellmacher, The Theory of Finite Groups, An Introduction, (Springer-Verlag, 2004). ISBN 0-387-40510-0. (Up to about page 100).
3. G.D. James and M. Liebeck, Representations and Characters of Groups (Second edition, Cambridge University Press, 2001). ISBN 0-521-00392-X

The following more sophisticated books are useful for reference and, in approach, may better represent the spirit of this course:

1. J I Alperin and Rowen B Bell, Groups and Representations, Graduate Texts in Mathematics 162 (Springer-Verlag, 1995). ISBN 0-387-94526-1
2. M J Collins, Representations and Characters of Finite Groups, Cambridge Studies in Advanced Mathematics 22, Cambridge University Press, 1990; reprinted p/b 2008, ISBN 978-0-521-06764-5, esp pp 48-63.

All these, and many other books on finite group theory and introductory character theory, can be found in most college libraries. You might also try the internet.