The Influence Of Weakly H-Subgroups And Weakly H^~*-Subgroups On The Structure Of Finite Groups

Let G be a finite group, H< G. H is called an H-subgroup of G if NG(H) ∩H9<H for all g ∈ G;H is called a weakly H-subgroup in G if there is a normal subgroup K of G such that G=HK and H ∩K is an H-subgroup in G; H is called a weakly H*-subgroup in G if there is a subgroup K of G such that G=HK and H ∩K is an H-subgroup in G. In the study of finite groups, using some of the properties of special subgroups to characterize the structure of finite groups is an important method. In this paper, by using the properties of weakly H-subgroup and weakly H*-subgroup of prime power order, we investigate the characterization of nilpotent, supersolvability and p-nilpotent of G, and we get some new characterizations of nilpotent, supersolvability and p-nilpotent of G. This paper is divided into two chaptersn. The first chapter, we mainly give the concept of weakly H-subgroup and weakly H*-subgroup, introduce the investigative background, the preliminary notions and some relevant known results, the main properties and relevant lemmas which are related to weakly H-subgroup and weakly H*-subgroup. In the second chapter, by using the weakly H-subgroup and weakly H*-subgroup of prime power order of G, we get some sufficient conditions for a finite group G to be nilpotent, p-nilpotent and supersolvable. We obtain some main results as follows:Theorem 2.1.1 Let G be a finite group and p the prime divisor of |G|. If all cyclic subgroups of order p and order 4 (p=2) of G are in H(G), then G is p-nilpotent.Theorem 2.1.2 Let G be a finite group and p the prime divisor of |G|, N be 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 when p=2 of N are in H(G), then G is p-nilpotent.Theorem 2.1.3 Let G be a finite group, N be a normal subgroup of G such that G/N is p-nilpotent. If every subgroup of order 4 of F*(N) is weakly H-subgroup in G. Then G is nilpotent if and only if every subgroup of order p of F*(N) is contained in Z∞(G).Theorem 2.1.4 Let G be a finite group and p(p≠ 2) the smallest prime divisor of |G|. P is a Sylow p-subgroup of G. If NG(P) is nilpotent and when P is not cyclic and Φ(P)=1, there exists a subgroup D of P with 1<|D|<|P|such that every subgroup of P of order |D| is weakly H-subgroup in G, then G is p-nilpotent.Theorem 2.2.1 Let G be a finite group and p the smallest prime divisor of |G|). If there exists an exchange subgroup P, where P is a Sylow p-subgroup of G, such that every subgroup of order p of P is weakly H*-subgroup in G, then G is p-nilpotent.Theorem 2.2.2 Let G be a finite group. All Frattini subgroup of p-subgroup of G is 1, where p is in π(G). If every subgroup of order p of G is weakly H*-subgroup in G, then G is p-nilpotent.