On the Frattini subgroup of a finite group

Abstract

We study the class of finite groups $G$ satisfying $\Phi (G/N)=\Phi (G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.

arXiv comment8 pages, no figures