Three books by Martin Isaacs — one on graduate algebra, one on finite group theory and one on undergraduate Euclidean geometry — have already been reviewed in this site. All three reviews are very favorable and note the quality of Isaacs’ writing. This excellent writing style is also very much in evidence in this book, which is, I believe, the first book that Isaacs ever wrote. Originally published by Academic Press in 1976, it now seems to be available both as a hardcover in the AMS/Chelsea series, as well as a less expensive (as of this writing, new copies were available at amazon.com for only twelve dollars) Dover paperback, thereby making it the only book, to my knowledge, that is simultaneously available from two different publishers. (I assume that the two versions are more or less identical, but, for the record, this review is of the Dover edition.)

Let’s begin with a quick summary of pertinent definitions and facts. A *representation* of a finite group G is simply a homomorphism ρ from G into the group GL_{n}(F) of nonsingular n × n matrices over a field F (or, what amounts to the same thing, a homomorphism into the group of nonsingular linear transformations of an n-dimensional vector space V over F). When F has positive characteristic, we speak of a *modular* representation. Representations can be studied from several points of view. In addition to thinking of them as homomorphisms, as defined above, we can also view V as a module over the ring F[G] (the group ring of G over F, defined as the set of all formal sums f_{1}g_{1} +… +f_{m}g_{m}, where the g_{i} are in G and the f_{i} are in F). The module structure here is given as (f_{1}g_{1} +… +f_{m}g_{m})v = f_{1}ρ(g_{1})v+… +f_{n} ρ(g_{n})v; conversely, it is easy to see that a module V over F[G] can be used to define a representation of G.

Yet a third approach to this theory is by means of *characters*: to a representation ρ we associate a function χ from G to F by defining χ(g) to be the trace of the matrix ρ(g). It is easy to see that the character does not determine the representation uniquely, since for any nonsingular matrix A the mapping g → A^{–1}ρ(g)A defines a new representation with the same character. However, this is (over the field of complex numbers) the extent of deviation allowed: if a complex representation ρ defines the character χ, then any other complex representation with character χ is obtained by conjugating ρ as above. Although not at all obvious, knowledge of the character χ gives a great deal of information about the group G, which of course is a major theme of this book.

As the title perhaps gives away, Isaacs approaches the theory of group representations from a viewpoint that is very much character-theoretic. He primarily works over the field of complex numbers, but several chapters also address the modular theory.

This is not a book for undergraduates (if you want one of those, check out Steinberg’s *Representation Theory of Finite Groups* for an approach that concentrates on the representation, or James and Liebeck’s *Representations and Characters of Finite Groups *for a somewhat more module-oriented look at things). Isaacs presupposes a good background in algebra, specifically knowledge of group theory at the level of a first year graduate course, or, in some places, perhaps even more: as the author states in the preface, knowledge of the transfer is useful in some places.

The writing style, as noted above, is very elegant, but quite concise: Isaacs thinks carefully about how he phrases things, and he does not waste words. Specific examples illustrating the concepts are given, but much more so in the early part of the text, less so in later chapters; an Appendix lists the character tables of eight specific finite groups. (The James and Liebeck book is also a particularly good source for specific character tables.) There are lots of exercises, most of which seemed to my non-expert eyes to be decidedly non-trivial; no solutions are provided.

The book begins with a chapter reviewing algebras and modules and culminating in the definition of an algebra representation. This first chapter, and the next five, comprise what Isaacs refers to in the preface as the elementary theory of characters, although people with just a casual prior acquaintance with the subject are likely to find a number of new concepts and ideas here. Even in these introductory chapters, the author’s expertise is evident; the proof presented of theorem 3.12 is due to him (and, independently, Glauberman).

These first six chapters take us quickly and efficiently through such topics as the basic definitions, the orthogonality relations, character tables, products of characters (this involves the tensor product), induced characters, applications to finite group theory (Burnside’s theorem that any group of order p^{a}q^{b} is solvable, described by Steinberg as the “first major triumph of representation theory”, is proved in chapter 3, for example) and much more. A number of these elementary results rely on the theory of algebraic numbers, the basics of which are developed as needed. My guess is that these chapters, comprising less than a hundred pages, would provide enough material for a one-semester graduate course on this theory.

The remaining chapters are of a somewhat more specialized nature, likely to appeal to people planning to do research in the area; some of the results appearing in this book were quite new and taken from the research literature. According to the bibliographic notes at the end of the book, for example, chapter 13 of the text is based almost entirely on a 1968 paper by Glauberman and most of chapter 12 is based on joint work of Isaacs and Passman. The final chapter of the book provides an introduction to Brauer’s theory of modular characters.

Just about all of the material in these later chapters was completely new to me, and this points out that there is always an inherent problem with having a non-specialist review a graduate-level book that is almost 40 years old: because research trends change, unsolved problems get solved, and new unsolved problems take center stage, only a person who is up to date on current research can give an authoritative opinion as to how well the book holds up as an entrée into current research. Lacking that expertise, I won’t even attempt to hazard an opinion. What I can say, though, is that this book is a genuine classic in the field, and anybody who is looking for a beautifully written introduction to the area will want to own it. The first five or six chapters alone are worth the relatively low price of admission.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.