The nilpotence class of the Frattini subgroup

Abstract

Ap-group of sufficiently large nilpotence class cannot occur as a normal subgroup contained in the Frattini subgroup of any finite group. The Frattini subgroup of a group of order Πpiαi with max αi at least 3, has nilpotence class at most 1/2 (max αi − 1). The Frattini subgroup of at-group is abelian. The occurrence of groups of orderp4 as normal subgroups contained in Frattini subgroups is investigated.