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 .