On the normalizers of p-nilpotency-residuals of all subgroups in a finite group

Abstract

Let G be a finite group. We define the subgroup N𝒩p(G) to be the intersection of the normalizers of the p-nilpotent residuals of all subgroups of G for a prime p. We first investigate the properties of N𝒩p(G), and then study the relationship between ⋂p‖G∣N𝒩p(G) and N𝒩(G), where N𝒩(G) is the intersection of the normalizers of the nilpotent residuals of all subgroups of G.