Non-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 C ? H or C ? H = 1 for each cyclic subgroup C of G.In this paper,it is given that.a criterion for some non-abelian p-groups such as A1-groups.A2-groups.metacyclic p-groups and groups of order p4 to be isolated in a finite p-group.The structure of finite p-group with an isolated A1-subgroup of index p(or P2)are determined.Janko's two results about isolated subgroups are improved,respectively.