Theory of Finite Simple Groups

Richard Brauer began his lecture at the 1954 International Congress of Mathematicians in Amsterdam with the following words: "The theory of groups of finite order has been rather in a state of stagnation in recent years. If I present here some investigations on groups of finite order, it is with the hope of raising new interest in the field."

In fact, much interest was raised by Brauer's talk because he began the formulation of a method to classify all finite simple groups. This program was enthusiastically followed to completion and resulted in the classification of simple groups that we know today. The "proof" of such classification, however, is spread over thousands of pages over several books and journals (some estimate the total length to be well above 10,000 pages; some call it "the enormous proof"). Unfortunately, there is much skepticism around this proof and a number of mathematicians doubt that these thousands of pages form a complete and gap-free demonstration of the facts claimed in the classification. Even more unfortunate is the fact that the announcement of the complete classification (which was, however, followed by the discovery of several gaps, now resolved) provoked a rapid decrease in the motivation of new researchers to enter this area.

The book under review, Theory of Finite Simple Groups , seems to perfectly fit the spirit of Brauer's words in the Congress of 1954. The volume constitutes an appealing and thorough introduction to the theory of finite simple groups, and also an introduction to the strategy laid down by Brauer and others with the goal of achieving a full classification. But, more importantly, the aim of the book is to present some recent methods that researchers hope will significantly simplify (and confirm!) the proof of the classification.

The book only assumes a one year graduate course in algebra, so the first chapter is dedicated to summarize the prerequisites from group theory (such as results on the presentation of groups). Chapter 2 goes over the basic theory of representations of finite groups (including Brauer's characterization of characters of finite groups). Chapter 3 concentrates on the proofs of Brauer's important theorems on p-blocks of finite groups.

In chapter 4, Michler uses representation theory techniques to prove the classical group order formulas of Brauer, Frobenius, Thompson and Suzuki, and presents a new formula (first proved by Michler) which is used later on to determine the order of all (known) sporadic simple groups having a unique conjugacy class of involutions. The chapter also features a new proof of a well-known criterion of non-simplicity, first proved by Brauer and Suzuki.

In chapters 5 and 6 the author discusses algorithms to determine all conjugacy classes and all values of all irreducible characters of finite permutation groups and finitely generated matrix groups over fields. The last five chapters contain many interesting applications of the theory produced in the previous chapters. In particular, the text presents proofs of the uniform existence and uniqueness for Janko's smallest sporadic group, the Higman-Sims group, the Harada group and the Thompson group. It is worth pointing out that the uniqueness of the Thompson group was a long standing open problem in the theory of finite simple groups, and the results in the last chapter of the book (due to Weller, Previtali and Michler) provide an answer to this problem.

The reader with some experience using the computer packages MAGMA and GAP will particularly benefit from this book because of the numerous algorithms and pieces of code included in the text. The book also mentions functions that have already been defined in one of the packages.

In summary, this volume is a very interesting, well-written and well-organized introduction to a beautiful area of mathematics, from a fresh point of view. The theory of finite simple groups deserves to capture (once again) the attention of many more researchers and Michler's book will greatly help graduate students in the study of these topics.

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.