# finiteness of an abelian topological group

Let A be an abelian (Hausdorff) topological group. Assume that

(1) the set of its torsion elements, and

(2) a finitely generated subgroup

are dense subsets of A.

My question: must A be finite?

(This is clearly true if A is discrete or if the f.g. subgroup is torsion).

EDIT. I'm more interested in compact groups.

## 1 Answer

Consider $$S^1$$, the multiplicative group of complex numbers of absolute value $$1$$. The torsion elements are roots of unity, which are dense in $$S^1$$. Also the subgroup generated by any non-torsion element, say $$\exp(2\pi it)$$ with $$t$$ irrationals, is dense in $$S^1$$.

Of course, $$S^1$$ is compact and infinite.

• Thanks! I should have thought about it. ... A friend of mine (years ahead in math) told me that I knew a counterexample but I should look at totally disconnected groups. Do you know if my question is true for t.d. groups? – J. Alegre Apr 2 '19 at 5:09
• If I'm not mistaken the product of Z/pZ over p answers it negatively. – J. Alegre Apr 2 '19 at 6:29