Is a finitely generated metrizable group discrete?

Question

The question is in the title. A countable locally compact Hausdorff group is discrete, so saying that a finitely generated metrizable group is locally compact would be enough.

What if the group is a subgroup of a compact metrizable group? An open or closed subspace of a compact Hausdorff space is locally compact, but here there is no hypothesis on opennes or closure.

Answer 1

The $p$-adic topology on $\mathbb{Z}$, having as a basis of neighborhoods the subgroups of $\mathbb{Z}$ of the form $p^n\mathbb{Z}$ ($p$ a prime) is metrizable and non discrete. The completion is the group (or ring) of $p$-adic integers, which is compact and metrizable.

Answer 2

No. Consider for instance the subgroup of $\mathbb{R}/\mathbb{Z}$ generated $a + \mathbb{Z}$ where $a$ is irrational, with the natural metric inherited from $\mathbb{R}/\mathbb{Z}$.

It is true though that every countable completely metrizable topological group is discrete, which follows from the more general fact that every countable (edit) homogeneous (/edit) complete metric space is discrete.