The Influence Of Some New Features Of Subgroups On The Structure Of Finite Groups

The relationship between the features of subgroups and the structure of finite groups has always been one of research focus in finite groups.This paper mainly consider-s the structures of finite groups from the aspect of developing the concept of weakly s-semipermutable subgroups,that is,we investigate the p-supersolvability,supersolv-ability and p-nilpotence of finite groups by using the nearly s-embedded and weakly s-supplemented subgroups respectively and expanding our partial results to formations of groups.To be specific,the first part of the main conclusion of this paper,we consider the nearly s-embedded subgroup of finite group.Firstly,we assume that every maximal subgroup of some Sylow p-subgorup of a group G is nearly s-embedded in G,then either G is p-supersolvable or G has a cyclic Sylow p-subgroup.Furthermore,we investigate the subnormal subgroup containing the generalized p-Fitting subgroup of G under the given conditions and obtain some sufficient and necessary conditions for a finite group G to be p-nilpotent or supersolvable.Secondly,we assume that the normalize of some Sylow subgroup of a finite group is p-nilpotent and characterize the p-nilpotency of finite group by using all of the maximal subgroups and part of maximal subgroups of some Sylow subgroup of finite groups respectively.Meanwhile,we illustrate the theoretical significance well with the help of GAP.Finally,we also considered the influence of some local embedding of subgroups on the structure of finite groups.That is,we assume that every maximal subgroup of the Sylow p-subgroup P of a group G is nearly s-embedded in NG(P)and P' is s-permutable in G.And we get some sufficient conditions for a group G to be p-nilpotent and also a elementary result for a factorized group to be p-nilpotent.The second part of the main conclusion of this paper,we consider the weakly s-supplemented subgroup of finite groups.We firstly obtain two sufficient conditions respectively for a finite group to be p-nilpotent by using all of maximal and minimal subgroups of the Sylow subgroup.And then we exactly characterize the p-nilpency of even order and odd order groups respectively.That is to say,we let D be the given subgroup of some Sylow p-subgroup P of G,and we investigate every subgroup of P with order |D| or p|D|.Finally,we expanding our results above to formations of groups.