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.