Fully inert subgroups of free Abelian groups

A subgroup H of an Abelian group G is called fully inert if (φH + H)/H is finite for every φ ∈ End(G). Fully inert subgroups of free Abelian groups are characterized. It is proved that H is fully inert in the free group G if and only if it is commensurable with nG for some n ≥ 0, that is, (H + nG)/H and (H + nG)/nG are both finite. From this fact we derive a more structural characterization of fully inert subgroups H of free groups G, in terms of the Ulm–Kaplansky invariants of G/H and the Hill–Megibben invariants of the exact sequence 0 → H → G → G/H → 0.