The Influence Of Semi-p-cover-avoiding Or SS-quasinormal Subgroups On The Structure Of Finite Groups

In this paper, by studying the semi-p-cover-avoiding and SS-quasinormal sub-groups, we obtain some results of the structure of finite groups, such as nilpotency, p-nilpotency, p-supersolvability. The full paper is divided into chapters:The first chapter, we introduce the investigative background, the preliminary no-tions, correlative theorems and lemmas.The second chapter, we use the method of or in subgroups by combining the properties of SS-quasinormal and semi-p-cover-avoiding of subgroups can still hold, and obtain some sufficient and necessary conditions for a finite group to be p-nilpotent, p-supersolvable. Some main results as follows:Theorem2.1.1Let F be a p-nilpotent saturated formation, p∈π(G) and with gcd(|G|,p-1)=1. Suppose N is a normal subgroup of a group G such that G/N is p-nilpotent. Then the following statements are equivalent:(a)G is p-nilpotent.(b)Ψp(N) is contained in ZF(G) and p=2, in addition, every subgroup T con-tained in Ψ4(N) either lies in ZF(G) or is Sp-CAP or SS-quasinormal subgroup of G and NG(T) is p-nilpotent.Theorem2.1.4Let p∈π(G) and G be a finite group with gcd(|G|,p-1)=1. Suppose N is a normal subgroup of a group G such that G/N is p-nilpotent, if every cyclic subgroup T of Ψ4(F*(N)) of order4is Sp-CAP or SS-quasinormal subgroup of G and NG(T) is p-nilpotent. Then the follow statements are equivalent:(a)G is p-nilpotent. (b)Every element x of Ψp(F*(N)) is contained in the hyper-center Z∞(G) of G.Theorem2.1.6Let F be a S-closed saturated formation containing Np and G be a group, p∈π(G). If N is a normal subgroup of G such that G/N∈F. Then the following statements are equivalent:(a)G∈F.(b) Every cyclic subgroup of Ψ(GF) of order4is Sp-CAP or SS-quasinormal subgroup of G and<x> lies in the F-hyper-center ZF(G) of G, for every element x∈Ψp(GF).Theorem2.1.7Let G be a finite group and p be a fixed prime number, F be a p-nilpotent saturated formation. If every subgroup of G of order p is contained in Zf(G) and p=2, in addition, every element y∈Ψ4(G),<y>is Sp-CAP or SS-quasinormal subgroup of G, or lies in Zj=(G). Then G is p-nilpotent.Theorem2.2.1Let G be a p-solvable group and p be the smallest prime divisor of|G|, p∈π(G). If G is A,4-free and there exists a normal subgroup H of G such that G/H is p-supersolvable, and all maximal subgroups of Fp(H)(the p-Fitting subgroup of H) are Sp-CAP or SS-quasinormal subgroups of G. Then G is p-supersolvable.Theorem2.2.2Let F be a saturated formation containing Up and p be an odd prime divisor of|G|with a solvable normal subgroup E such that G/E∈F. If all maximal subgroups of every noncyclic Sylow subgroup P of Fp(E) are Sp-CAP or SS-quasinormal subgroups of G. Then G∈F.