Here is a question I posted some months ago in Math.SE, and t.b. mentioned to the following question by Florent MARTIN which is somehow related to my question;

Let $G$ be a compact Hausdorff topological group, and let $H$ be a torsion-free group satisfying the ascending condition, i.e. there are no infinite strictly ascending chains $H_1 < H_2<...$ of subgroups of $H.$

Prove that there is no non-trivial homomorphism of $G$ into $H.$

Note that, no topology is considered on $H$ and "homomorphism" simply means "group homomorphism."