The Power Subgroups And The Structure Of Groups

Let G be a group, H a subgroup of G, If H =< gn|g G > for a non-negative integer n, then we call H a power subgroups of G and denote H by Gn.In this dissertation, by investigating the power subgroup of a group,we have proved the following results:Theorem 1.1 If G is a nilpotent torsion group, then G satisfies |G : Gp} < for every p (G) if, and only if, every Sylow p- subgroup Gp of G is an extension of the center by a finite p- group, and also satisfies | (GP) : ( (GP))p| < .Theorem 1.2 Let G be a locally nilpotent torsion FC- group. If G satisfies |G : Gp| < for every p (G) , then every Sylow p- subgroup Gp of G satisfies |GP : (GP)| < , and | (GP) : ( (Cp))p| < .Theorem 1.3 Let G be a soluble p- group with expG < . If G satisfies |G : Gp| < , then G is a finite group.Theorem 1.4 Let G be a hypercenter group with expG < . If G satisfies |G : Gp| < for every p (G), then G is a finite group.Theorem 1.5 Let G be a locally finite group, then for every proper sugroup H of G, H satisfies |H : Hp| < for every p (H) and H Hp if, and only if, G is a locally nilpotent group and every Sylow p- subgroup Gp of G is finite.Theorem 2.1 Let G be a p- group with |G : Gp| = , but for every proper subgroup H of G satisfying |H : Hp| < , then p 2,3; If p 5, then for each non-central element g G - Gp , one has G =< x >G Gp.Theorem 2.2 Let G be a p- group with |G : Gp| = , but for every normal subgroup N of G satisfying N G and |G/N : (G/N}P| < ,then p 2,3; If p 5, then for every element g G - Gp, x has infinitly many conjugate elements.Theorem 3.1 Let G be a finite group, If G has only 4 subgroups which are not power subgroups of G, then G Z3 Z3.