[Here](https://math.stackexchange.com/questions/60355/no-non-trivial-homomorphism-to-a-group) is a question I posted some months ago in Math.SE, and t.b. mentioned to the following [question](https://mathoverflow.net/questions/80966/) 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." 



No non-trivial homomorphism to a group
There are no non-trivial homomorphisms from a compact group to a torsion-free finitely generated group by the theorem of Nikolov and Segal quoted in the answer by Andreas to Florent's question (mentioned by the OP above). Since ascending chain condition on subgroups implies finite generation, this answers the question. 


Nb. Please upvote Andreas's answer if you like this one. 


