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