Some results on variations on the norm of finite groups

Let G be a finite group and \(N_{\Omega }(G)\) $N_{\mathrm{\Omega }}(G)$ be the intersection of the normalizers of all subgroups belonging to the set \(\Omega (G),\) $\mathrm{\Omega }(G),$ where \(\Omega (G)\) $\mathrm{\Omega }(G)$ is a set of all subgroups of G which have some theoretical group property. In this paper, we show that \(N_{\Omega }(G)= Z_{\infty }(G)\) $N_{\mathrm{\Omega }}(G)=Z_{\infty }(G)$ if \(\Omega (G)\) $\mathrm{\Omega }(G)$ 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 weakly normal subgroups of G; (v) the set of all NE-subgroups of G.