IRREDUCIBLE MODULES OF SOLVABLE GROUPS ARE ALGEBRAIC

Abstract

Work of Brauer and Dade has shown that if P, a p-Sylow subgroup of G where G is a finite group is cyclic, then rather explicit connections can be found between the ordinary characters of G and those of NG(P). Efforts to generalize these results so far have been unsuccessful. One of the many difficulties standing in the way of generalizations is a lack of knowledge about indecomposable modules for noncyclic p-groups at characteristic p. In the case of a cyclic p-group, the indecomposables are |P| in number and are given by certain Jordan block matrices. If P is noncyclic then no such easy answer is available. The initial observation to be made is not to consider all indecomposable modules. The idea is to understand the indecomposable summands of V|p where V is some irreducible k[G]-module.