Problems in the theory of block induction

This dissertation is devoted to the study of block induction in the modular representation theory of finite groups.;We introduce the concept of strong covering of blocks in order to analyze extended block induction, which is the weakest among the various types of block induction. We establish a theorem on p-local characterizations of strong covering. As an application of this result to the study of the behavior of principal blocks, we prove one direction of Brauer's Third Main Theorem under extended block induction in p-solvable groups. This result is not true in general. We also give an example which shows that the other direction of Brauer's Third Main Theorem does not hold even in solvable groups. We find a class of infinitely many counterexamples to the transitivity of extended block induction. Nevertheless, for some special cases, we do obtain some positive results. By means of the computer software MAGMA, we find a counterexample to the transitivity of strong covering in a solvable group. In order to give a characterization of block induction in Brauer's sense, we consider the set of "centralizable" characters, which includes all the multiples of virtually irreducible characters and all the multiples of characters of height zero. We prove a theorem which characterizes Brauer's induction in terms of centralizable characters. As corollaries, we obtain some known results in block theory.