Cambridge Studies in Advanced Mathematics, 11. Cambridge etc.: Cambridge University Press. X, 178 p. £20.00; $ 29.95 (1986).
The book gives an introduction to the (modular) representation theory of finite groups, using a module-theoretic approach. This is one of basically three approaches to the theory, the others being character- theoretic (in the style of Brauer) and ring theoretic (Puig et al.). To get the full picture one has to combine all three approaches. Consequently a full treatise of the most important topics has to be of considerable length. One advantage of this nice book is its brevity. Although it has only 178 pages it leads the reader (almost) from the first principles to one of the high points of the subject, the cyclic theory.
In chapter I (Semisimple modules) and Chapter II (Projective modules) the author presents proofs of some basic facts about A-modules, where A is a finite-dimensional algebra over a field of characteristic p, or more specially a group algebra. In Chapter III (Modules and subgroups) we find for instance an exposition of the theory of relative projective modules, vertices, sources and Green correspondence. Chapter IV (Blocks) contains the following sections: Defect Groups. Brauer correspondence. Canonical module. Subpairs. The final chapter covers the cyclic theory (in appr. 50 pages). This theory has been the subject of a considerable amount of research since Dade’s paper started it all, and it is very difficult to write an exposition of it. The theory is complicated, no matter how you do it. The author has obviously taken particular care in writing this chapter and the result is probably the most accessible version yet. (Needless to say that there is more to be said about the cyclic theory than what is included here.)
Alperin’s book is strongly recommendable. It has something to offer to both specialists and non-specialists. It is the obvious choice as the first book to read on the topic, but the readers should not forget its limitations in approach. The book will hopefully awake more interest in a very fruitful area of research.