A Course in Finite Group Representation Theory

To put in a single phrase, representation theory aims to understand the abstract structure of a group by looking at its varying shadows on matrix groups. Or, equivalently, by looking at its actions on linear spaces or other geometrical or combinatorial objects.

The book under review, aimed at graduate students with more than a basic background on groups, rings, fields and linear algebra, provides an introduction to finite groups representations accessible not only to future specialists but also to students whose interests lie in other areas of mathematics. The choice of topics reflects the stated goals.

The book content could be divided in three parts. The first part, chapters 1 to 5, is the author’s take on the ordinary or semisimple case, that is, complex linear representations and characters of finite groups. In these 95 pages, the topics covered range from Maskche’s and Wedderburn’s theorems and Schur’s lemma to character tables and Burnside’s \(p^aq^b\)-theorem, ending with induction and restriction of representations and Mackey’s and Clifford’s theorems. The second part, chapters 6 to 9, starts the study of the modular case, that is, the case when the characteristic of the ground field divides the order of the group. The crucial difference is that in this case the corresponding modules are not necessarily semisimple. The description of these modules is the objective of chapters 7 and 8, first in the case when the corresponding group algebra is finite dimensional, and then in the general case. Chapter 9 considers some rationality questions when the ground field is not algebraically closed. The main results are the existence of a splitting field extension of finite degree and the construction and main properties of the decomposition map.

The third part, chapters 10 to 12, goes deeper into the study of modular representations, starting with the theory of Brauer characters, which in characteristic \(p\) recover some of the properties of ordinary characters in characteristic \(0\). Chapter 11 considers more general indecomposable modules and the structure of the corresponding endomorphism rings, and chapter 12 gives an introduction to block theory.

Each of the twelve chapters comes with sets of exercises, emphasizing the textbook nature of the book. One appendix summarizes some facts on (discrete) valuations and a second appendix collects character tables for some finite groups. This is a well-written and motivated book, with carefully chosen topics, examples and exercises to engage the reader, making it suitable in the classroom or for self-study.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.