Some results on the norm of finite groups

Let G be a finite group and NΩ(G) be the intersection of the normalizers of all subgroups belonging to the set Ω(G), where Ω(G) is a set of all subgroups of G which have some theoretical group property. In this paper, we show that NΩ(G)= Z∞(G) if Ω(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 H-subgroups of G; (v) the set of all weakly normal subgroups of G; (vi) the set of all $NE$-subgroups of G.