Subgroups of monothetic groups

It is shown that every separable abelian topological group is isomorphic with a topological subgroup of a monothetic group (that is, a topological group with a single topological generator). In particular, every separable metrizable abelian group embeds into a metrizable monothetic group. More generally, we describe all topological groups that can be embedded into monothetic groups: they are exactly the abelian topological groups of weight less than or equal to c covered by countably many translations of every non-empty open subset.