Representation theory has very strong interplay with group structure. This is particularly true for finite solvable groups G, because their chief factors are irreducible modules for G over fields of prime order. In this monograph, we present some topics and problems arising in the representation theory of solvable groups. In particular, we study modules over finite fields, yet give applications to ordinary and Brauer characters of solvable groups.

It is not our intent to develop representation theory from scratch, but rather to discuss techniques and problems in current research. On the other hand, we wish that the manuscript be accessible to a reasonably wide group of people, including advanced graduate students, working in group theory. We refer to two basic references, namely:

1. [Hu] B. Huppert, “Endliche Gruppen I” and

2. [Is] I. M. Isaacs, “Character Theory of Finite Groups”.

We believe that readers fairly familiar with these texts should have little problem reading the manuscript. We do also quote some material from the first chapter appearing in the sequel to [Hu], namely Chapter VII of “Finite Groups II” by B. Huppert and N. Blackburn [HB]. That chapter is entitled “Elements of General Representation Theory”. Many of the results from these sources for which we have frequent use are presented (generally without proof) in Chapter 0, “Preliminaries”. Since we present some applications to block theory, we state and/or prove several related results in Chapter 0.