The Influence Of SS-Semipermutable Subgroups On The Structure Of Finite Groups

In the investigative process of finite groups, the use of some properties of some special subgroups (for instance:maximal subgroups, Sylow p-subgroups) to characterize and de-scribe the structure of finite group and research the related properties of group is a significant method and pathway. In this thesis, by using S-semipermutability or SS-semipermutability of maximal subgroup of Sylow p-subgroup, we investigate the characterization of p-nilpotency and supersolvability of G, and we obtain some new sufficient conditions of p-nilpotency and supersolvability of G. The thesis is divided into two chapters according to contents. In the first chapter, we mainly introduce the concept of S-semipermutable and SS-semipertable subgroup, then introduce the investigative background, the preliminary definitions and some relevant known results, the main properties and correlative lemmas. The second chapter is divided into two sections:In the first section, by using the S-semipermutable group, we dis-cuss the characterization of supersolvability and p-nilpotency of G. In the second section, using the SS-semipermutable subgroup of prime power order of G to discuss p-nilpotency of a finite group G. Some sufficient conditions for a finite group G to be p-nilpotent were obtained, which are generalization of some results in correlative papers. We obtain some mainly results as follows:Theorem 2.1.1 Let G be a p-solvable group and P be a Sylow p-subgroup of G, where p is a prime divisor of|G|. If all maximal subgroup of P are S-semipermutable in NG(P), P’is S-semipermutable in G, then G is p-supersolvable.Theorem 2.1.2 Let G be a finite group and P be a Sylow p-subgroup of G, where p is the smallest prime divisor of|G|. If all maximal subgroup of P are S-semipermutable in NG(P), P’is S-semipermutable in G, then G is p-nilpotent.Theorem 2.1.4 Let G be a finite group. If there exists P E Sylp(G) such that all maximal subgroup of P is S-semipermutable in NG(P) for an arbitrary prime divisor p of |G|. And P’is S-semipermutable in G, then G is supersolvable.Theorem 2.1.6 Let G be a finite group and H be a normal subgroup of G such that G/H is p-supersolvable. If all maximal subgroup of P are S-semipermutable in NG(P), where P is a Sylow p-subgroup of F*(H), then G is p-supersolvable.Theorem 2.1.7 Let F be a saturated formation containing the class of supersolvable groups U and G be a finite group, N(?) G such that G/N ∈ F. If all maximal subgroup of P is S-semipermutable in NG(P) for an arbitrary Sylow p-subgroup P oi H, and P’is S-semipermutable in G, then G E F.Theorem 2.2.1 Let G be a finite group and let P be a Sylow p-subgroup of G, where p is a prime divisor of|G|with (|G|,p-1)= 1. If every cyclic subgroup of P with order p or 4 (if p= 2) is SS-semipermutable in G, then G is p-nilpotent group. Theorem 2.2.3 Let G be a finite group, Let p be an odd prime dividing the order of a group G and P be a Sylow p-subgroup of G. If P is quaternion-free and all minimal subgroups of D(G) ∩ P are SS-semipermutable in G, then G is p-nilpotent.Theorem 2.2.4 Let G be a finite group, and P be a Sylow p-subgroup of G, p be an odd prime dividing the order of G. Suppose that NG(P) is a p-nilpotent group and there exists a subgroup D of P with 1<|D|<|P|such that every subgroup H of P with order |D| is SS-semipermutable in G. Then G is a p-nilpotent group.Theorem 2.2.5 Let G be a finite group, N be a normal subgroup of G such that G/N is a p-nilpotent group, and P be a Sylow p-subgroup of N, where p is an odd prime dividing the order of D. Suppose that NG(P) is a p-nilpotent group and there exists a subgroup D of P with 1<|D|<|P|such that every subgroup H of P with order|H|=|D|is SS-semipermutable in G. Then G is a p-nilpotent group.