# No non-trivial homomorphism to a group

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."

-
If $G$ is finite, it's false. So I guess you should take $G$ infinite. Do you want to show that there is no injective homomorphism $G \to H$? Because then you just need to show that $G$ fails to have the ascending condition. Hint: $G$ is uncountable. If that isn't what you mean, then take $G$ profinite and let $H$ be a finite quotient. Or am I missing something? –  Richard Kent Dec 20 '11 at 1:22
Doesn't the answer to Florent's question answer yours? The acc implies finitely generated and by the answer to that question all fg images are finite. –  Benjamin Steinberg Dec 20 '11 at 1:23
Richard, he said H is torsion-free so in particular not finite. –  Benjamin Steinberg Dec 20 '11 at 1:24
Oh, thanks. Missed it. –  Richard Kent Dec 20 '11 at 3:16