On Weakly SS-Quasinormally Embedded Subgroups And S-Semipermutable Subgroups

Let G be a finite group and let H be a subgroup of G.H is said to be quasinormal in G if H permutes with every subgroup of G;H is said to be S-quasinormal in G if H permutes with every Sylow subgroup of G;H is said to be S-quainormally embedded in G if every Sylow subgroup of H is also a Sylow subgroup of some S-quasinorrmal subgroup of G;H is said to be a weakly S-quainormally subgroup of G if there exists T??G such that G = HT and H ?T ?HsG,where HsG is the maximal S-quainormally subgroup of G which contains H;H is said to be a weakly SS-quasinormally embedded subgroup of G if there exists a normal subgroup T of G such that HT?G and every Sylow subgroup of H ? T is also a Sylow subgroup of S-quasinormal subgroup of G;H is said to be a semipermutable subgroup of G if it is permutable with every subgroup K of G with(|H|,|K|)=1;H is said to be a S-semipermutable subgroup of G if it is permutable with every Sylow p-subgroup of G with(p,|K|)=1.In the investigation of finite groups,using order of group or properties of subgroup or properties of element to protray the structure and discuss properties of finite groups is a main direction and a common approach.In this paper,throughing the weakening and extension of the normality to discuss the properties of G,we get some new characterizations on G when the subgroup of G is weakly SS-quasinormally embedded or S-semipermutable.This article is divided into two chapters according to the content:The first chapter,we mainly introduce the investigative background of the weakly SS-quasinormally embedded subgroup and the S-semipermutable subgroup,and give the basic definitions and some relevant known results,the main properties and correlative lemmas which are related to this paper.The second chapter,by using the weakly SS-quasinormally embeddedness and the S-semipermution to research the structure of groups.The main results are as follows:(1)Let G be a p-solvable group,p is a prime factor dividing |G|.If every maximal subgroup of Fp(G)contained in Op'(G)is weakly SS-quasinormally embedded in G,then G is-supersolvable.(2)Let G be a finite group,p||G| and(|G|,p-1)= 1,P ?Sylp(G).If every maximal subgroup of P is weakly SS-quasinormally embedded in G,then G is p-nilpotent.(3)Let G be a p-solvable group,H(?)G,p is a prime factor dividing |G|,G/H be a supersolvable group.If the maximal subgroups of Sylow p-subgroup of H is S-semipermutable in G,then G is p-supersolvable.(4)Let G be a finite group,p|G| and p is a prime,H is a nomal subgroup of G and G/H is p-supersolvable.If the maximal subgroups of Sylow p-subgroup of H is S-semipermutable in G,then G is p-supersolvable.(5)Let G be a finite group,H(?)G7 G/H is supersolvable,every minimal subgroup of F(H)and cyclic subgroup of order four is S-semipermutable in G,then G is supersolvable.