Nearly S-Semipermutable Subgroups Of Finite Groups

It is an effective way to study the structure of a finite group by using the permutable subgroups on which a lot of famous theorems were set up. The concept of a nearly s-semipermutable subgroups is introduced in this paper. Let G be a finite group and H be a subgroup of G.H is said to be nearly s-semipermutable in G if there is a subgroup T of G such that G=HT and H∩T is s-semipermutable in T. In this paper, We investigate the influence of the nearly s-semipermutable subgroups on the structure of a finite group:in particular, we give some new conditions for a group to be p-nilpotent or supersoluble and some of them are generalized to a given formation.The paper consists of the following four sections:In section one:we introduce some backgrounds of our research.In section two:we introduce some symbols. basic concepts, and lemma as used in this paper.In section three:with restrictions on nearly s-semipermutable subgroups of maximal subgroups or minimal subgroups of some Sylow subgroup of G. some sufficient and necessary conditions of supersoluble groups and a given formation are obtained.In section four:with assumtion on nearly s-semipermutable subgroups of maxi-mal subgroups or minimal subgroups of some Sylow subgroup of G, some sufficient conditions of p-nilpotent groups are obtained.