topological groups - Dense cyclic subgroup

First, it is clear the group has to be abelian. Now, if you assume that $G$ is locally compact, then by the classification you can decompose $G$ as $G={mathbb R}^n times H$ where
$H$ has a compact open subgroup. Clearly, there can be no ${mathbb R}^n$ factor, so $G$ has a
compact open subgroup. Now, suppose $G$ is itself compact and topologically generated by $g$. Then any character $chi$ in the dual of $G$ vanishing on $g$ will be identically zero. So, the map
$chi mapsto chi (g)$ is injective, hence the dual is a subgroup of $U(1)$. Conversely, you can also see that if $Gamma$ is a subgroup of $U(1)$ (considered with the discrete topology) then the dual of $Gamma$ has a dense cyclic subgroup. By taking various subgroups you can, for instance, get the $p$-adic integers, or the n-torus.