ℱ𝒦-norms of finite groups

Abstract

[Formula: see text] always represents a finite group. [Formula: see text] is a formation meeting the condition [Formula: see text] where [Formula: see text] and [Formula: see text], respectively, represent the class of nilpotent groups and abelian groups. [Formula: see text] is a subgroup-closed formation. We call [Formula: see text] the [Formula: see text]-norm of [Formula: see text] if [Formula: see text] is the intersection of the normalizers of [Formula: see text] for all the subgroups [Formula: see text] of [Formula: see text] satisfying [Formula: see text] We mainly demonstrate that [Formula: see text] We also show that [Formula: see text] if and only if [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text] generated by three elements. Furthermore, we introduce the [Formula: see text]-groups and [Formula: see text]-[Formula: see text]-groups.