The Influence Of Nearly SS-Embedded Subgroups And Weaklly Quasinormal Subgroup On The Structure Of Finite Groups

Let G be a finite group, a subgroup H of G and a subgroup K of G is called permutation, if HK=KH. A subgroup H of G is said to be S-quasinormal in G if HP= PH for every Sylow subgroup P of G. A subgroup H of G is said to be S-quasinormally embedded in G, if every Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. A subgroup H of G is said to be nearly SS-embedded in G if G has an S-quasinormal subgroup T such that.HT is S-quasinormal in G and H∩T< HseG, where H∩T≤HseG is the subgroup generated by all those subgroups of H which are S-quasinormally embedded in G. A subgroup H of G is said to be weakly quasinormal subgroup M of G, there exist x E G such that HMx= MXH.The method used in this thesis to investigate the structure and the property of a finite group is to describe a finite group according to its subgroups, and it is the most com-monly used method in group theory. The thesis takes full advantage of the property of S-quasinormal subgroup,S-quasinormally embedded subgroup, nearly SS-embedded sub-group and weakly quasinormal subgroup by researchist of the theory of finite group, and proceed more research to the finite group.The thesis is divided into two chapters. In the first chapter, we mainly introduce the background and give some definitions and some important lemmas. In the second chap-ter, according to the property of nearly SS-embedded subgroup and weakly quasinormal subgroup research finite group, some new results are obtained as follows:Theorem 2.1.1 Let G be a finite group, where p is the smallest prime dividing|G|, let P be a Sylow p-subgroup of G,and (|G|, (p-1)(p2-1)… (pn-1))= 1, where 1≤ n≤7. If every n-maximal subgroup of P is nearly SS-embedded in G, then G is solvable.Theorem 2.1.2 Let G be a finite group, where p is the prime dividing|G|, let P be a Sylow p-subgroup of G, and (|G|, (p-1)(p2-1) … (pn-1))= 1, where 1≤ n≤ 7. If every n-maximal subgroup of P is nearly SS-embedded in G, then G is p-nilpotent.Theorem 2.1.3 Let G be a finite group, where p is the smallest prime dividing|G| and p ≠ 2, let P be a Sylow p-subgroup of G. If Nq(P) is p-nilpotent and every n-maximal subgroup of P is nearly SS-embedded in G, where 1≤ n≤ 2, then G is solvable.Theorem 2.2.1 Let p be a prime and F a saturated formation containing all p-nilpotent groups. Suppose that G is a finite group with (|G|, (p-1)(p2-1)… (pn-1))= 1, where 1≤ n≤ 7, Then G ∈F if and only if G has a normal subgroup H such that G/H ∈F and H has a Sylow p-subgroup P such that every n-maximal subgroup (if exists) of P is weakly quasinormal subgroup in G.Theorem 2.2.2 Let G be a finite group, where p is the prime dividing|G|, (|G|, (p-1)(p2-1) … (pn-1))= l,where 1≤ n≤7, then G is p-nilpotent, assume that one of the following conditions is satisfied:(1) either p>2 or n≥2, for every order pn of subgroup of G is weakly quasinormal in G.(2) p= 2 and n= 1, for every subgroup of order 2 and 4 is weakly quasinormal in G.