Some results on variations on the norm of finite groups

Abstract

AbstractLet G be a finite group and $$N_{\Omega }(G)$$ N Ω ( G ) be the intersection of the normalizers of all subgroups belonging to the set $$\Omega (G),$$ Ω ( G ) , where $$\Omega (G)$$ Ω ( 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 Ω ( G ) = Z ∞ ( G ) if $$\Omega (G)$$ 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 weakly normal subgroups of G; (v) the set of all NE-subgroups of G.