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 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."
No non-trivial homomorphism to a group
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$\begingroup$
group-theory
topological-groups
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1A somewhat related question was asked and [answered on MO](http://mathoverflow.net/questions/80966/), recently. I haven't looked at the paper mentioned in the answer, but maybe it contains something useful for this question. – 2011-11-15
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0@t.b:The given thread solves the problem. Thank you. – 2011-12-25
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0Ehsanmo, could you then answer your own question and accept the answer so this question doesn't show up in unanswered questions? Thank you. – 2012-03-05
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0@ymar: Yes, sure. – 2012-03-05
1 Answers
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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!