Some results on the norm of finite groups

Abstract

Let $G$ be a finite group and $N_{\Omega }(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega (G),$ where $\Omega (G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this paper, we show that $N_{\Omega }(G)=Z_{\infty }(G)$ if $\Omega (G)$ is one of the following: (i) the set of all self-normalizing subgroups of $G$; (ii) the set of all subgroups of $G$ satisfying the subnormalizer condition in $G$; (iii) the set of all pronormal subgroups of $G$; (iv) the set of all $\mathcal{H}$-subgroups of $G$; (v) the set of all weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$.