The Influence Of SS-quasinormality And Weakly SS-semipermutability On The Structures Of Finite Groups

For a long time,the effect of the properties of subgroup on the structure of finite groups is one of the important topics of finite group theory.In this paper,we mainly discuss the structure of finite group based on the fact that all exactly n-minimal subgroup are SS-quasinormal,and the structure of finite group whose some subgroup are weakly SS-semipermutable.In the first chapter,we introduce the research background.In the second chapter,we introduce some basic concepts and lemmas.The third chapter is divided into two parts:In section 3.1,we study G whose some exactly n-minimal subgroup is SS-quasinormal.In this process,we focus on the counterexample of minimal order and the propertie of SS-quasinormal subgroup,We get the structure of G whose some exactly n-minimal subgroup is SS-quasinormal of G.The results are as follows:Theorem 3.1.1 Let F be a normal nilpotent subgroup of G and n be a positive integer.Suppose that every subgroup H of F with ω(H)=n is SS-quasinormal in G.Then the following statements are ture:(1)Suppose thatΦ(F)=1 and ω(F)≥ n+1.Then all subgroups of F are SS-quasinormal in G.(2)Suppose that E be a minimal G-invariant subgroup of F.Then ω(E)≤n and if in addition ω(F)≥n+1,then either ω(E)=1 or ω(E)≤n-1.Theorem 3.1.2 Let F be a normal nilpotent subgroup of G,and P be a Sylow p-subgroup of F where p is an odd prime.If all minimal subgroups of F are SS-quasinormal in G,or if ω(F)≥ 3 and all subgroups D of F with ω(D)= 2 are SS-quasinormal in G.Then P is G-supersolvable.Theorem 3.1.3 Let F be a normal nilpotent subgroup of G,and R be a Sylow 2-subgroup of F.Suppose that all cyclic subgroups of F with order 2 or 4 are SS-quasinormal in G,or that ω(F)>3 and all subgroups D of F with ω(D)=2 are SS-quasinormal in G.Then G/CG(R)is a 2-group,and R is G-supersolvable.In section 3.2,we discuss the maximal subgroup subgroup of Sylow subgroup is weakly SS-semipermutable,and use the counterexample of minimal order to confirm,we get the structure of finite group.The results are as follows:Theorem 3.2.1 Let G of odd order,p be the smallest prime factor of |G|,P be a Sylow p-subgroup of G.If every maximal subgroup of P is weakly SS-semipermutable in G,and P’ is s-permutable in G.Then G is p-nilpotent.Theorem 3.2.2 Let G be a p-solvable group for a prime p and P a Sylow p-subgroup of G.Suppose that every maximal subgroup of P is weakly SS-semipermutable in G.Then G is p-supersolvable.In the fourth chapter is the summary and prospect,summary the research in this article,and put forward we next work and research direction.