Researches On Problems Related To P-nilpotent Groups

The theory of finite groups occupies a prominent position both in theory and practical application.Its permutation groups,solvable and non-solvable groups,nilpotent groups,and group representation theory are all important research objects.In the investigation of finite groups,by using some properties of some special subgroups to characterize the structure of finite groups is an important method.In this thesis,by using S-semipermutablity,SS-quasinormality and related properties of HC-subgroups of some special subgroups(maximal subgroups,minimal subgroups,Sylow subgroups)of G,we obtain some sufficient conditions for a finite group to be p-nilpotent,and improves some related results.The main research content of this thesis is discussed from the following three parts.Firstly,We obtain two sufficient conditions for finite groups to be p-nilpotent groups by using the weakly S-semipermutablity of maximal subgroups and S-semipermutablity of derived subgroup of Sylow subgroup.for instance,Let G be a finite group and p is the smallest prime dividing|G|,P∈Syl_p(G),If every maximal subgroup of P is weakly S-semipermutablity inN _G(P)andP’is S-semipermutablity in G,then G is p-nilpotent.Secondly,We mainly use the SS-quasinormal of cyclic subgroups of order 4 and prime subgroups of G,we obtain some sufficient conditions for a finite group to be p-nilpotent.for instance,Let G be a finite group,p is a prime factor of|G|,If every p-subgroup of G is contained in Z_∞(G),every cyclic subgroup of order 4 of G is SS-quasinormal in G.Then G is p-nilpotent.Lastly,We mainly apply the properties of HC-subgroups and obtain two sufficient condition for a group to be p-nilpotent.for instance,Let G be a finite group,and p is the smallest prime factor of the order of G,and there is an exchange p-subgroup in G,and if each minimal subgroup of P is a HC-subgroup in G,then G is p-nilpotent.Finally,the research content of the subject is summarized and the research results of this paper are further prospected.