Some Sufficient Conditions Of Solvable Groups And P -nilpotent Groups

In the investigation of finite groups, using the permutability of normal subgroups to charac-terize the properties of group can obtain some profound results.Two subgroups H and K of a group G are said to permute if HK= KH. A subgroup H of group G is said to be S-quasinormal in G if H permute with every Sylow subgroup of G, which was introduced by Kegel and was well in-vestigated by many authors, for example, see Asaad, Yangming Li and Yanming Wang. Recently, Ballester-Bolincher and Pedraza-Aguilera have extended this concept to S-quasinormal embedded in G if every Sylow subgroup P of H, there is a S-quasinormal subgroup K in G such that P is aslo a Sylow subgoup of K.In this paper, we will introduce a new way to study another generalization of S-quasinormal subgroup. Recall that a supplement of H to G is a subgroup B such that G= AB. Based on these concepts, Shirong Li gave the definition of SS-quasinormal subgroup.A subgroup A of G is said to be an SS-quasinormal subgroup of G if there is a supplement B of A to G such that A permutes with every Sylow subgroup of B. The purpose of this paper is, by using the SS-quasinormal subgroups, to study the properties of finite groups (such as solvability, p-supersolvability, p-nilpotency). It is divided into two chapters. In the first chapter, we introduce the investigative backgroud of this paper and some preliminary notions, properties and correlative lemmas, therorems. In the second chapter, we obtain some sufficient conditions of a group which is solvable group, supersolvable group,p-nilpotent group, and we also prove the following results:Theorem 2.1.1 Let p be a prime dividing the order of a group, P be Sylow p-subgroup of G and (|G|, p-1)= 1. If P is SS-quasinormal in G, then G is p-solvable. Particular, if every subgroup of G is SS-quasinormal in G, then G is solvable.In the first part, by using the SS-quasinomal of maximal subgroups, some sufficient condi-tions of solvable groups were obtained.Theorem 2.1.3 Let M be a maximal subgroup of G. If every Sylow subgroup is SS-quasinormal in G, then G is solvable.Theorem 2.1.4 Let p be the smallest prime dividing the order of a group G and P a Sylow p-subgroup, A is a fixed maximal subgroup of P. If A is SS-quasinormal in G, then G is solvable. Theorem 2.1.5 Let M be a maximal subgroup of even order. If M is not p-nilpotent but whose proper subgroups are all p-nilpotent and every Sylow p-subgroup is SS-qusinormal in G, then G is solvable.In the second part, we consider the noncyclic Sylow subgroup of G, obtain some sufficient condtions for supersovability of a finite group.Theorem 2.1.8 Let G be a group. If, for every prime p dividing the order of G and P a noncyclic Sylow p-subgoup of G, every member of some fixed Md(P) is SS-quasinormal in G, then G is supersolvable.Theorem 2.1.10 Let H be a normal subgroup of G such that G/H is supersolvable. If every maximal subgroup of noncyclic Sylow subgroup of N is SS-quasinormal in G, then G is supersov-able.Theorem 2.1.12 Let F be a saturated formation containing U and let G be a group. Then the two following statements are equivalent:(1)G∈F;(2) There exists a normal subgroup H of G such that G/H∈F, and if every maximal subgroup of noncyclic sylow subgroup of H is SS-quasinormal in G.Theorem 2.1.14 Let F be a saturated formation containing U and let G be a group. Then the two following statements are equivalent:(1)G∈F;(2) There exists a normal A-subgroup H of G such that G/H∈F, and every maximal sub-group of noncyclic Sylow subgroup of H is SS-quasinormal in G.Theorem 2.1.16 Let G be a QCLT group and G2 a Sylow 2-subgroup of G. If the commutator subgroup G'2 of G2 is SS-quasinormal in G, then G is supersolvable.Theorem 2.1.17 Let G be a QCLT group and G2 a Sylow 2-subgroup of G. If every maximal subgoup of G2 is SS-quasinormal in G, then G is supersolvable.In the third part, we consider the relationship between subgroups of prime order and SS-quasinormal subgroups, we obtain some main results as follows:Theorem 2.1.21 Let G=AB, where A and B are proper subgroups of G and G' is nilpotent. If all subgoups of prime order of both A and B are SS-quasinormal in G, then G is supersolvable.Theorem 2.1.24 Let F be a saturated formation containing U and H with the Sylow tower a normal subgroup of a group G such that G/H∈F. If, for every Sylow subgroup Q of H, the cyclic subgroups of prime order or order 4 of Q are SS-quasinormal in NG(Q), then G∈F.In the fourth part, using SS-quasinormal of some special subgroups, we obtain some sufficient conditions for a group to be p-nilpotent.Theorem 2.2.1 Let p be the smallest prime dividing|G| and P a Sylow p-subgroup of G of exponent pn, where n≥1. Suppose that all members of the set{H|H≤P, H'=1, exp H=pn) are S S-quasinormal in G, then G is nilpotent.Theorem 2.2.2 Let G be a finite group and p a prime dividing the order of G such that (|G|, p-1)=1. If there exists a normal subgroup N of G such that G/N is p-nilpotent and every subgroup of prime and order 4(if p=2) of G is SS-quasinomal in G, then G is p-nilpotent.Theorem 2.2.5 Let G be a finite group and p a prime dividing the order of G. Suppose (|G|, p-1)=1 and G is A4-free. If there exists a normal subgroup N of G such that G/N is p-nilpotent and every subgroup of order p2 of every Sylow p-subgroup of N is SS-quasinormal in G, then G is p-nilpotent.Theorem 2.2.9 Let p be the smallest prime dividing the order of a group G. If every subgroup of order p is SS-quasinormal in G and the Sylow p-subgroup of G is abelian, then G is p-nilpotent.