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: no topology is considered on $H$ and "homomorphism" simply means "group homomorphism."

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    $\begingroup$ A somewhat related question was asked and answered on MO, recently. I haven't looked at the paper mentioned in the answer, but maybe it contains something useful for this question. $\endgroup$ – t.b. Nov 15 '11 at 22:53
  • $\begingroup$ @t.b:The given thread solves the problem. Thank you. $\endgroup$ – Ehsan M. Kermani Dec 25 '11 at 4:46
  • $\begingroup$ Ehsanmo, could you then answer your own question and accept the answer so this question doesn't show up in unanswered questions? Thank you. $\endgroup$ – user23211 Mar 5 '12 at 9:16
  • $\begingroup$ @ymar: Yes, sure. $\endgroup$ – Ehsan M. Kermani Mar 5 '12 at 9:26

It is easy to see that such an $H$ is finitely generated and the rest follows from Nikolov-Segal theorem.

I wonder if there is a non high-tech way to prove it though!


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