The norm of p-decomposable residuals of all subgroups in a finite group

Abstract

For a finite group [Formula: see text], we define the subgroup [Formula: see text] to be the intersection of the normalizers of the [Formula: see text]-decomposable residuals of all subgroups of [Formula: see text] for a prime [Formula: see text]. Set [Formula: see text]. Define [Formula: see text] for [Formula: see text] By [Formula: see text] denote the terminal term of the ascending series. It is proved that [Formula: see text] if and only if the [Formula: see text]-decomposable residual [Formula: see text] is nilpotent. Furthermore, if the index of [Formula: see text] is co-prime with [Formula: see text], then [Formula: see text] is [Formula: see text]-solvable with [Formula: see text].