Fully inert subgroups of free Abelian groups

A subgroup H of an Abelian group G is called fully inert if ( ϕ H + 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 n G for some n ≥ 0 , that is, ( H + n G ) / H and ( H + n G ) / n G 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 .