If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the subspace topology) infinite cyclic subgroup?

An easy necessary condition comes from the usual decomposition theorem that any LCA group $G$ can be expressed as $G = \mathbb{R}^n \times H$ for some $H$ which contains a compact-open subgroup, so we know that any such $G$ must contain a compact-open subgroup, but this is obviously nowhere near sufficient.

If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the subspace topology) infinite cyclic subgroup?

An easy necessary condition comes from the usual decomposition theorem that any LCA group $G$ can be expressed as $G = \mathbb{R}^n \times H$ for some $H$ which contains a compact-open subgroup, so we know that any such $G$ must contain a compact-open subgroup, but this is obviously nowhere near sufficient.

EDIT: Another sufficient condition is that $G$ is topologically torsion, that is, every element is contained in a compact subgroup.