Fully inert subgroups of Abelian p-groups

A subgroup H of an Abelian group G is said to be fully inert in G, if for every endomorphism ϕ of G  , the factor group (H+ϕ(H))/H(H+ϕ(H))/H is finite. This notion arises in the study of the dynamical properties of endomorphisms (entropy). The principal result of this work is that fully inert subgroups of direct sums of cyclic p-groups are commensurable with fully invariant subgroups of the direct sum.