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Let $ f: G \to H $ be a group homomorphism. Suppose that the induced map $ F: \text{Hom}(H,H) \to \text{Hom}(G,H) $ defined by $ F(\phi) \stackrel{\text{def}}{=} \phi \circ f $ is a bijection. Show that if $ G $ is abelian, then so is $ H $.

I'm wondering if there is a fancy categorical proof of this theorem.

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    The proof I know you show f(G) is contained in Z(H), so that all inner automorphisms in H agree with the identity on f(G). Hence they are all the identity and H is abelian. But it looks like something nicer should exist.2011-08-15
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    What I'm hoping (and I have no reason to) is that Yoneda lemma somehow applies, because I'd like a better understanding of it.2011-08-15
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    It's quite easy to show that the condition on the hom-set map implies $f$ is an epimorphism. But at the moment I'm not seeing a good abstract-nonsense proof that epimorphisms transport abelian group structure...2011-08-15
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    An epimorphism in the category of groups is necessarily surjective...2011-08-15
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    @Keenan: Yes, but that's cheating. :p I was wondering about the following: Suppose $G$ is a group object in a category $\mathbf{C}$; given an epimorphism $f : G \to H$, does it follow that $H$ is also a group object, in a way making $f$ an internal homomorphism? As is well-known, an abelian group is simply a group object in the category of groups...2011-08-15
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    Perhaps I'm being dumb, but how can you tell $f$ is an epimorphism? Also, I don't understand your comment about an abelian group being the same as a group object in the category of groups.2011-08-15
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    @Keenan: Oops, it seems I have assumed something extra. I was thinking about the fact that $f : G \to H$ is an epimorphism if and only if $\textrm{Hom}(f, K) : \textrm{Hom}(H, K) \to \textrm{Hom}(G, K)$ is injective for every $K$. As for group objects in the category of groups, essentially, the Eckmann–Hilton argument shows that any such must be an abelian group.2011-08-15
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    @Zhen: there are tons of surjective functions from a group $G$ to a set $X$ which are not homomorphisms for any group structure on $X$...2011-10-23

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