Generalized Normality Of Subgroups And Characterization Of Properties Of Finite Groups

In this paper,we discuss the generalized normality of some subgroups of finite groups to research the properties of finite groups.We discuss three kinds of problems:First:G is a finite group,a subgroup H of G is said n-embedded in G if G has a normal subgroup T such that T n H<HsG and HT=HG.We consider the Sylow subgroups,maximal subgroups of Sylow subgroups,ect,which are n-embedded in G,we obtain some solvable results of finite groups.Second:G is a finite group,R is a subset of G,a subgroup H of G will be called R-subnormal subgroup if H is a subnormal subgroup of(H,R).By researching R-subnormality of Sylow subgroups,we get some results about p-nilpotent,nilpotent,supersolvable properties of finite groups.Third:We introduce a new definition of weakly(?)h-supplemented subgroups.Let(?)be a class of groups,a subgroup H of G is said to be weakly(?)h-supplemented in G if there exists a S-quasinormal subgroup T of G such that HT is a Hall subgroup of G and(H(?)T)HG/HG≤Z∞(?)(G/HG),where Z∞(?)(G/HG)is(?)-hypercenter of G/HG.We use the properties of weakly(?)h-supplemented of maximal subgroups of Sylow subgroups,2-maximal subgroups of Sylow subgroups of G,ect,and obtain some results about p-nilpotence,supersolvability of finite groups.