The Influence Of C-Normal Or SS-Semipermutable Subgroups On The Structure Of Finite Groups

Let G be a finite group. A subgroup H of G is called SS-semipermutable in G if there is a supplement B of H to G such that H permutes with every Sylow p-subgroup of B with (p,|H|)= 1. Here, H is also said to be an SS-semipermutable subgroup of G(supplement-Sylow-semipernutable subgroup).In the investigation of finite groups, using some properties of subgroups to charac-terize the structure of finite group G is a effective method. The purpose of this paper is to study the influence of C-normal subgroups and SS-semipermutable subgroups on the structure of finite groups such as p-nilpotency and supersolvability. The pa-per is divided into two chapters. In the first chapter, we introduce the investigative background, the preliminary notions, correlative lemmas and Theorems. In the sec-ond chapter, we use the properties of the C-normal subgroups and SS-semipermutable subgroups to investigate the structure of finite groups, and obtain some sufficient con-ditions for a finite group to be p-nilpotent and supersolvable. We obtain some main results as follows:Theorem 2.1.1 Let G be a finite group and p be a prime dividing the order of G with (|G|,p-1)= 1,P∈Sylp(G). If every maximal subgroup of P is C-normal or SS-semipermutable in G, Then G/Op(G) is p-nilpotent.Theorem 2.1.2 Let G be a finite group and p be an odd prime dividing the order of G, P∈Sylp(G). If NG(P) is p-nilpotent and every maximal subgroup of P is C-normal or SS-semipermutable in G, Then G is p-nilpotent.Theorem 2.1.5 Let G be a finite group and p be a prime dividing the order of G with (|G|,p - 1)= 1. If there exists a normal subgroup N of G such that G/N is p-nilpotent and every maximal subgroup of every Sylow subgroup of N is C-normal or SS-semipermutable in G. Then G is p-nilpotent.Theorem 2.2.1 Let G be a finite group with a normal subgroup N such that G/N is supersolvable. If every maximal subgroups of noncyclic Sylow subgroup of N is C-normal or SS-semipermutable in G, then G is supersolvable.Theorem 2.2.3 Let F be a saturated formation containing u and suppose that G is a finite group with a solvable normal subgroup N such that G/N∈F. If every maxi-mal subgroup of noncyclic Sylow subgroup of F(N) is C-normal or SS-semipermutable in G, then G∈F.Theorem 2.2.5 Let G be a finite group with a normal subgroup H such that G/H is supersolvable. If all minimal subgroups and every cyclic subgroup of order 4 of H are C-normal or SS-semipermutable in G, then G is supersolvable.Theorem 2.2.7 Let G be a finite group with a normal subgroup H such that G/H is supersolvable. If all minimal subgroups of H are C-normal or SS-semipermutable in G, and the Sylow 2-subgroup of H is abel, then G is supersolvable.Theorem 2.2.9 Let G be a finite group with a solvable normal subgroup H such that G/H is supersolvable. If all minimal subgroups and every cyclic subgroup of order 4 of F(H) are C-normal or SS-semipermutable in G, then G is supersolvable.Theorem 2.2.13 Let G be a finite group with a solvable normal subgroup H such that G/H is supersolvable. If all minimal subgroups of H are C-normal or SS-semipermutable in G, and the Sylow 2-subgroup of F(H) is abel, then G is supersolv-able.