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What would be an example of two non-isomorphic groups $G$ and $G'$ such that for all groups $H$, there are as many homomorphisms $H\to G$ as homomorphisms $H\to G'$?

By the Yoneda lemma, if $\hom(-,G)$ is naturally isomorphic to $\hom(-,G')$, then $G$ is isomorphic to $G'$. My question aims at finding an example that shows that it doesn't suffice to just assume that $\hom(H,G)$ and $\hom(H,G')$ are isomorphic for all $H$.

Alex Provost
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user907204
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  • What are your thoughts on the problem? For example, do you think your groups can have the same order? Or be finite/infinite? – user1729 Mar 27 '21 at 16:30
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    I see why it has been closed, but it is an interesting question. I would look for an example of infinite groups $G$ in which the number of homomorphisms from a group $H$ to $G$ is either $1$ or infinite. – Derek Holt Mar 27 '21 at 16:58
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    @Derek I agree it's interesting. My vote-to-close was because of a complete lack of attempt or effort or anything. (I think for infinite groups it's not too hard, but the finite case seems more subtle.) – user1729 Mar 27 '21 at 17:05
  • Thanks, added context! Hope the question is alright now. – user907204 Mar 27 '21 at 17:54
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    @diracdeltafunk Yes! Thanks. – user907204 Mar 27 '21 at 18:20
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    For the question as stated, (pairs of) non-cyclic free groups are counter-examples as they each contain each other. If you restricted to finite groups then this question would not be covered by the linked question, and would be quite interesting. – user1729 Mar 27 '21 at 19:21

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