On the Frattini subgroup of a finite group

TL;DR

This paper investigates a specific property of finite groups related to the Frattini subgroup, extending known results from soluble groups to all finite groups and addressing a longstanding open question.

Contribution

It extends and amplifies a theorem about the Frattini subgroup property from soluble groups to all finite groups and confirms a long-standing conjecture about complements of normal subgroups.

Findings

Extended theorem to all finite groups

Confirmed class of groups with normal subgroup complements is subnormally closed

Provided new insights into the structure of finite groups

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.