The Influence Of CSS-Quasinormal Subgroups On The Structure Of Finite Groups

Let G be a finite group. A subgroup If of a group G is said to be quasinormal in G, if HK=KH for every subgroup K of G. A subgroup if of a group G is said to be SS-quasinormal in G if there exists a supplement M of H in G such that if permutes with every Sylow subgroup of M. A subgroup H of a group G is said to be CSS-quasinormal subgroup in G if there exists a quasinormal subgroup T of G such that G=HT and if H∩T is SS-quasinormal in G.In the investigation of finite groups, by using some properties of some special sub-groups to characterize the structure of finite group is a main method. In this paper, by using the properties of CSS-quasinormal subgroups of prime power order, we investigate the characterization of supersolvability and p-nilpotency of G, and we get some new char-acterizations of supersolvability and p-nilpotency of G. The full thesis is divided into two chapters according to contents: In the first chapter, we mainly give the new concept of CSS-quasinormal subgroup, introduce the investigative background, the preliminary notions and some relevant known results, the main properties and correlative lemmas which are related to CSS-quasinormal subgroups. In the second chapter, by using the CSS-quasinormal sub-group of prime power order of G, we get some sufficient conditions for a finite group G to be p-nilpotent and supersolvable. We obtain some main results as follows:Lemma1.2.2Let G be a finite group, and H≤K≤G. Then:(1) If H is CSS-quasinormal in G, then H is CSS-quasinormal in K;(2) If H is SS-quasinormal in G, then H is CSS-quasinormal in G;(3) If if is a normal subgroup of G, then K/H is CSS-quasinormal in G/H if and only if K is CSS-quasinormal in G;(4) If if is a normal subgroup of G, M is CSS-quasinormal in G with (|H|,|M|)=1, then HM/H is CSS-quasinormal in G/H. Theorem2.1.1Let G be a finite group and p the smallest prime divisor of|G|, G is A4-free. If all cyclic subgroups of order p and order4(p=2) of G are CSS-quasinormal in G, then G is p-nilpotent.Theorem2.1.2Let G be a finite group and p a prime divisor of|G|such that (|G|, p-1)=1, G is A4-free. If there exists a normal subgroup N of G such that G/N is p-nilpotent and P is a Sylow p-subgroup of N, all cyclic subgroups of order p and order4(p=2) of P are GSS-quasinormal in G, then G is p-nilpotent.Theorem2.1.5Let G be a finite group and p the smallest prime divisor of|G|, 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 p2of P is GSS-quasinormal in G, where P is a Sylow p-subgroup of N, then G is p-nilpotent.Theorem2.1.10Let G be a finite group and p a prime divisor of|G|such that (|G|,p2-1)=1. If there exists a normal subgroup N of G such that G/N is p-nilpotent and every subgroup of order p2of N is GSS-quasinormal in G, then G is p-nilpotent.Theorem2.2.1Let F be the class of groups with Sylow tower of supersolvable type and p the smallest prime divisor of|G|, G is A4-free. If there exists a normal subgroup N of G such that G/N∈F, all cyclic subgroups of prime order and order4(p=2) of P are GSS-quasinormal in G, where P is a Sylow p-subgroup of N, then G∈G.Theorem2.2.2Let F be the class of groups with Sylow tower of supersolvable type and p a prime divisor of|G|, G is A4-free. If there exists a normal subgroup N of G such that G/N∈F, every subgroup of order p2of P∩OP(G) is CSS-quasinormal in G, where P is a Sylow p-subgroup of N, then G∈F.