The representation theory of finite groups can be approached from several points of view: One can use the classical group-theory (or character-theory) approach, keeping the group properties readily at hand, or use ring theory, or use module-theory, with emphasis either on the associated rings or algebras or the corresponding category of modules. It needs no elaboration to say that all of these approaches are necessary and even required to understand the representations of finite groups.
Since a (linear) representation of a (finite) group is a vector space V, over a field k, together with a linear action of G, then the representation theory of a finite group G naturally splits in two cases: The semisimple (ordinary) case, when the characteristic of the field k does not divide the order of the group G, and the modular case, when characteristic of the field divides the order of the group. The ordinary case is relatively easy, since the group algebra k[G] splits into irreducible components forcing all representations to decompose as direct sums of irreducible representations. In the modular case the structure of the representations is rather more complicated.
The ordinary case has been well understood since the pioneering work by Frobenius, Schur and Burnside and its translation into the language of algebras and modules by E. Noether and her school. There are many good expositions of this case, either in the language of modules or in the language of characters, for example in M. Isaacs’ Character Theory of Finite Groups (Academic, 1976, reprinted by Dover in 1994 and more recently by AMS-Chelsea in 2006).
The modular case has been treated, from the character-theory point of view, in several monographs, for example in D. Goldschmidt’s Lectures on Character Theory (Publish or Perish, 1980) or in G. Navarro’s Characters and Blocks of Finite Groups (Cambridge, 1998). The character-theory approach does not allow the development of Green’s theory of indecomposable modules. For this, there are other books, for example J. Alperin’s Local Representation Theory (Cambridge, 1986) uses the module-theory approach and P. Landrock’s Finite Group Algebras and their Modules (Cambridge, 1983) uses ring-theoretic methods.
The classical books The Representation Theory of Finite Groups (North-Holland, 1982) by W. Feit, and C. Curtis and I. Reiner’s Representation Theory of Finite Groups and Associative Algebras (Interscience, 1962, reprinted in by AMS-Chelsea in 2006) treat the modular representation of finite groups using the three approaches mentioned before.
The book under review is an introduction to the modular representation theory of finite groups with a somehow balanced approach to the subject. It develops enough ring and module theoretic methods to treat Green’s theory of indecomposable representations, the Grothendieck group of the group algebra k[G], its ring structure, the Burnside ring and Clifford theory. With these tools, the author proves the main induction theorems. Moreover, the book also introduces the main character-theoretic methods in a natural fashion: From the Brauer character and its main properties, to blocks, defect groups and the Brauer correspondence.
Assuming a small amount of the ordinary representation theory, the book is almost self-contained. It has the lightness of a gentle-paced lecture course and could be used with profit for an introduction to the methods of representation theory of finite groups, either in a formal course or for self-study. For a formal course the instructor must provide more exercises, since the book comes with very few of them.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.