On NS*-permutable Subgroups Of Finite Groups

Let H be a subgroup of a finite group G. H is said to be.NS-permutable in G, if for all primes p such that (p,|H|)= 1, and for every subgroup K of G containing H, the normalizer NK (H) contains some Sylow p-subgroup of K. H is said to be NS*-permutable in G, if there exists a quasinormal subgroup K of G such that G= HK and H n K is NSS-permutable in G. In this thesis, we mainly investigate the influence of NS*-permutable subgroups on the structure of finite groups, then use its related properties to explore some sufficient conditions about supersolvability and p-nilpotency of G. The thesis is divided into two chapters according to contents:In the first chapter, we mainly introduce the preliminary notions, known results, and correlative lemmas which are needed in the sequal. In the second chapter, we use the characters of NS*-permutable subgroups to discuss the supersolvability and p-nilpotency of finite groups. We obtain some main results as follows:Theorem 2.1.1 Let G be a finite group and p an odd prime divisor of |G|, P is a Sylow p-subgroup of G. If every maximal subgroup of P is NS-permutable in G and Nq(P) is a p-nilpotent group, then G is p-nilpotent.Theorem 2.1.3 Let G be a finite group and p a prime divisor of |G|, N is a normal subgroup of G such that G/N is p-nilpotent, P is a Sylow p-subgroup of N. If (|G|,p-1)= 1 and every maximal subgroup of P is NS-permutable in G, then G is p-nilpotent.Theorem 2.1.4 Let G be a finite group and p a prime divisor of |G| such that (|G|,p-1)= 1, P is a Sylow p-subgroup of G. If every minimal subgroup of P is NS-pexmutable in G and when p= 2, P is quaternion-free, then G is p-nilpotent.Theorem 2.1.6 Let G be a finite group and p a prime divisor of |G| such that (|G|,p-1)= 1, N is a normal subgroup of G such that G/N is p-nilpotent, P is a Sylow p-subgroup of N. If every minimal subgroup of P is NS-permutable in G and when p= 2, P is quaternion-free, then G is p-nilpotent.Theorem 2.1.8 Let G be a finite group and p a prime divisor of|G|such that (|G|,p-1)= 1. If every cyclic subgroups of order p and order 4 (p= 2) of G is NS-permutable in G, then G is p-nilpotent.Theorem 2.1.10 Let G be a finite group and p a prime divisor of|G|such that (|G|,p-1)= 1, N is a normal subgroup of G such that G/N is p-nilpotent. If all cyclic subgroups of order p and order 4 (p=2) of N are NS’-permutable in G, then G is p-nilpotent.Theorem 2.2.1 Let G be a finite group and G2 is a Sylow 2-subgroup of G. If G satisfies the permutable condition and every maximal subgroup of G2 is.NS-permutable in G, then G is p-nilpotent.Theorem 2.2.2 Let G be a finite group N is a normal subgroup of G such that G/N is p-nilpotent. If every maximal subgroup of Sylow subgroup of N is NS-permutable in G, then G is p-nilpotent.Theorem 2.3.1 Let G be a finite group and p a prime divisor of |G|such that (|G|,p-1)= 1, N is a normal subgroup of G such that G/N is p-nilpotent. If every cyclic subgroups of order p and order 4{p=2) of N are NS*-permutable in G, then G is p-nilpotent.Theorem 2.3.6 Let G be a finite group and p a prime divisor of |G|. If every subgroup of order p of G is contained in Z∞(G), and every cyclic subgroups of order 4(p=2) of G are NS*-permutable in G, then G is p-nilpotent.Theorem 2.3.8 Let G be a finite group and p a prime divisor of |G|, N is a normal subgroup of G such that G/N is p-nilpotent. If every subgroup of order p of N is contained in Z∞(G), and every cyclic subgroups of order 4(p=2) of N are NS*-permutable in G, then G is p-nilpotent.Theorem 2.3.10 Let G be a finite group and p the smallest prime divisor of |G|, N is a normal subgroup of G such that G/N is p-nilpotent, P is a Sylow p-subgroup of N. If every subgroup of order p2 of P are NS*-permutable in G and G is quaternion-free, then G is p-nilpotent.