The Influence Of Some Subgroups On The Structure Of Finite Groups

In recent years,it has been an interesting topic to use generalized normalities of subgroups and their quoticnt groups to describe properties of a group.Many investigators in this filed introuce some new subgroups such as S-seminormality,pronormal subgroup,weak c-normality,c~*-normality.And give some new criteria for the structrue of finite groups.For example,Yanming Wang in[10]introduced c-normality and gave some new criteria for the sovability and supersolvability of groups.In[13], Huanquan Wei introduced c~*-normality and gave some new criteria for the nilpotenty and supersolvability of groups.In this present paper,we study the influence of some subgroups on the structrue of finite groups.The first part mainly introudces the influence of s-seminormal subgroup on the structure of finite groups,and gives some corollaies about those subgroups that conditions are much strong compared to the s-seminormal subgroup.For example:Theorem 3.2 Let G be a finite group.Suppose that N(?)G,G/N is nilpotent. If≤Z_∞(G) for every element x of N with a prime order p when p∈π(N),and all cyclic subgroups of N with order 4 are s-seminormal in G when 2∈π(N).Then G is nilpotent.The second part mainly introduces the influence of a special subgroup called pronormal subgroup on the structrue of finite groups,and some theorems are obtained. For example:Theorem 4.3 Let G be a finite group.Suppose that N(?)G,G/N is nilpotent .If every element x of N with a prime order p is a weakly left Engel element of G whenp∈π(N),and all cyclic subgroups of N with order 4 are pronormal subgroup in G when 2∈π(N).Then G is nilpotent.The third part mainly introduces the influence of weak c-normal subgroup on the structure of finite groups,as well as has obtained some results about c-normal and s-normal on the structure of finite groups.For example:Theorem 5.5 Let G be a finite group.Suppose that every element x of G with a prime order p is a weakly left Engel element of N_G(x),all cyclic subgroups of G with order 4 and are weakly c-normal in G when 2∈π(G).Then G is nilpotent.The last part mainly introduces the influence of c~*-normal subgroup on the structrue of finite groups,and gives some new criteria for the nilpotentcy and super-solvability of finite groups.For example:Theorem 6.5 Let G be a finite group.Suppose that every element x of G with a prime order p is a weakly left Engel element of N_G(x),all cyclic subgroups of G with order 4 and are c~*-normal in G when 2∈π(G).Then G is nilpotent.Theorem 6.6 Let G be a finite group.Suppose that is c~*-normal in G for every element x of G with prime order or order 4.Then G is supersolvable.