Which infinite abelian groups admit an almost maximally almost-periodic group topology?

A topological group G is said to be almost maximally almost-periodic if its von Neumann radical n ( G ) is non-trivial, but finite. In this paper, we prove that every abelian group with an infinite torsion subgroup admits a (Hausdorff) almost maximally almost-periodic group topology. Some open problems are also formulated.