  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 $\mathscr{H}$-subgroups of $G$; (v) the set of all
weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$.
