Abelian P-group Which Can Be Acted As A Proper Isolated Subgroup Of A Finite P-group

Let G be a group.We say that a subgroup H of G is isolated in G if G≤H or C ∩ H = 1 for each cyclic subgroup C of G.In this paper,it is given that some necessary condition for an abelian p-group to be isolated in a finite p-group and a criterion for some abelian p-groups to be isolated in a finite p-group.Some equivalent condition for non-normal abelian(cyclic)subgroups to be isolated in a finite p-group are obtained.The structure of finite p-groups wih a proper isolated cyclic(two-generator abelian)subgroup are determined,respectively.