Characterization of almost maximally almost-periodic groups

Abstract

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