Principal blocks and p-radical groups

Abstract

Let G be a finite group and k a field of characteristic p > 0. In this paper, we obtain several equivalent conditions to determine whether the principal block B 0 of a finite p-solvable group G is p-radical, which means that B 0 has the property that e 0(k P ) G is semisimple as a kG-module, where P is a Sylow p-subgroup of G, k P is the trivial kP-module, (k P ) G is the induced module, and e 0 is the block idempotent of B 0. We also give the complete classification of a finite p-solvable group G which has not more than three simple B 0-modules where B 0 is p-radical.