  For any countable subgroup $H$ of an unbounded Abelian group $G$ there is a
complete Hausdorff group topology $\tau$ such that $H$ is the von Neumann
radical of $(G,\tau)$. In particular, we obtain the positive answer to
Comfort's question: any unbounded countable Abelian group admits a complete
Hausdorff minimally almost periodic (MinAP) group topology. A bounded infinite
Abelian group admits a MinAP group topology if and only if all its leading
Ulm-Kaplansky invariants are infinite. If, in addition, $G$ is countably
infinite, a MinAP group topology can be chosen to be complete.
