Kernel systems on finite groups

Abstract

We introduce a notion of kernel systems on finite groups: roughly speaking, a kernel system on the finite group $G$ consists in the data of a pseudo-Frobenius kernel in each maximal solvable subgroup of $G$, subject to certain natural conditions. In particular, each finite $CA$-group can be equipped with a canonical kernel system. We succeed in determining all finite groups with kernel system that also possess a Hall $p'$-subgroup for some prime factor $p$ of their order; this generalizes a previous result of ours (Communications in Algebra 18(3), 1990, pp. 833-838). Remarkable is the fact that we make no a priori abelianness hypothesis on the Sylow subgroups.