Characterization of almost maximally almost-periodic groups

Let G be an Abelian group. We prove that a group G admits a Hausdorff group topology τ such that the von Neumann radical n (G, τ) of (G, τ) is non-trivial and finite iff G has a non-trivial finite subgroup. If G is a topological group, then n (n (G)) ≠ n (G) if and only if n (G) is not dually embedded. In particular, n (n (Z, τ)) = n (Z, τ) for any Hausdorff group topology τ on Z.