On S-conditionally Semipermutable Subgroups Of Finite Groups And P-supersolubility

Let G be a finite group and H a subgroup of G.H is said to be S-semipermutable in G if it permutes with every Sylow p-subgroup P of G with(p,|H|) = 1;H is said to be S-conditionally semipermutable in G if there is x∈G such that HT~x =T_XH if(|H|,|T|)=1 for any Sylow subgroup T of G. In this paper,we investigate the influence of some S-conditionally semipermutable subgroups on the structure of finite groups,and generalize some recent results.Theorem 1 Let G be a finite group.G is p-supersoluble group if and only if there exists a p-soluble normal subgroup N of G such that G/N is a p-supersoluble group and every maximal subgroup of Sylow p-subgroup of N is S-conditionally semipermutable in G.Theorem 2 Let G be a finite group.G is p-supersoluble group if and only if there exists a p-soluble normal subgroup N of G such that G/N is a p-supersoluble group and every cycle p-subgroup of N is S-conditionally semipermutable in G.Theorem 3 Let G be a p-soluble and finite group.G is a p-supersoluble group if and only if there exists a normal subgroup of G such that G/N is a p-supersoluble group and every maximal subgroup of Sylow p-group of N is S-conditionally semipermutable in G or has a p-supersoluble supplement in G.