On the Intersection of the Normalizers of the $\boldsymbol{\cal{F}}$-Residuals of Subgroups of a Finite Group

Abstract

In this paper we give criteria for a finite group to belong to a formation. As applications, recent theorems of Li, Shen, Shi and Qian are generalized. Let G  be a finite group, $\cal F$ a formation and p  a prime. Let $D_{\mathcal {F}}(G)$ be the intersection of the normalizers of the $\cal F$-residuals of all subgroups of G, and let $D_{\mathcal {F}}^{p}(G)$ be the intersection of the normalizers of $(H^{\cal F}O_{p'}(G))$ for all subgroups H of G. We then define $D_{\mathcal F}^{0}(G)=D_{\mathcal F, p}^{~0}(G)=1$ and $D_{\mathcal F}^{i+1}(G)/D_{\mathcal F}^{i}(G)=D_{\mathcal F}(G/D_{\mathcal F}^{i}(G))$, $D_{\mathcal F, p}^{i+1}(G)/D_{\mathcal F, p}^{~i}(G)=D_{\mathcal F, p}(G/D_{\mathcal F, p}^{~i}(G))$. Let $D_{\mathcal {F}}^{\infty}(G)$ and $D_{\mathcal {F}, p}^{~\infty}(G)$ denote the terminal member of the ascending series of $D_{\mathcal F}^{i}(G)$ and $D_{\mathcal F, p}^{~i}(G)$ respectively. In this paper we prove that under certain hypotheses, the the $\cal F$-residual $G^{\cal F}$ is nilpotent (respectively,p-nilpotent) if and only if $G=D_{\mathcal {F}}^{\infty}(G)$ (respectively, $G=D_{\mathcal {F}, p}^{~\infty}(G)$). Further more, if the formation $\cal F$ is either the class of all nilpotent groups or the class of all abelian groups, then $G^{\cal F}$ is p-nilpotent if and only if and only if every cyclic subgroup of G order p and 4 (if p = 2) is contained in $D_{\mathcal {F}, p}^{~\infty}(G)$.