This book is concerned with the structure of group algebras of finite groups over fields of characteristic p dividing the order of the group, or closely related rings such as rings of algebraic integers and in particular their p-adic completions, as well as modules, and homomorphisms between them, of such group algebras.

Our principal aim has been to present some of the more recent ideas which have enriched and improved this beautiful theory that owes so much to Richard Brauer. In other words, we wish to account for a major part of what could be described as the post-Brauer period. The reader will find that once we get started, the majority of our proofs have not appeared before in any textbooks, and as far as Chapters II and III are concerned, a number of results and proofs which have not appeared before at all are included.

We do not at any stage restrict ourselves to particular methods, be they ring theoretic, character theoretic, etc. In each case we have attempted to present a proof or an approach which distinguishes itself in one way or another perhaps by being fast, elegant, illuminating, or with promising potentials for further advancement, or possibly all of this at the same time. (We are well aware of the fact that the reader may not always agree this has been achieved (unless of course he or she recognizes his or her own proof!)) One point though that has been important to us is to demonstrate the strong connection to cohomology which undoubtedly will be strengthened in the years to come.