The Influence Of Some Special Subgroups On The Structure Of Finite Groups

Let G be a finite group,H the subgroup of G.H is said to be weakly s-semipermutable in G if there is a subgroup T of G such that G=HT and H∩T≤H_SSG,where H_SSG is the subgroup of H generated by all those subgroups of H which are s-semipermutable in G.H is said to be M-supplemented in G if there is a subgroup B of G such that G=HB and H₁B is a proper subgroup of G for any maximal subgroup H₁ of H.In this paper,we study the relation between the weakly s-semipermutable subgroups or M-supplemented subgroups and the structure of the finite groups.The main results are the following:1.Let G be a group with a normal subgroup K such that G/K is supersolovable.If every cyclic subgroup of K of prime order or order 4 is weakly s-semipermutable in G,then G is supersolovable.2.Let G be a group with a normal subgroup K such that G/K is nilpotent. Suppose that every cyclic subgroup of K of order 4 is weakly s-semipermutable in G.Then G is nilpotent if and only ifχlies in the hypercenter Z_∞(G) of G for every elementχof K of prime order.3.Let G be a group with a normal subgroup K such that G/K is nilpotent. Suppose that every cyclic subgroup of K of order 4 is M-supplemented in G.Then G is nilpotent if and only ifχlies in the hypercenter Z_∞(G) of G for every elementχof K of prime order.