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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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I'm actually going to answer a slightly different question, which is the one I think I really wanted to ask: is this question motivated by category theory? I first saw this question on an old qualifying exam, and it seemed like there was more to it. In fact, I was told last week that it's related to the idea of localization. A good source of information on this concept can be found here (it wouldn't let me direct link to the pdf, but it's the paper number 101 called localizations on that page). The qualifying exam question can now be phrased: abelian groups are closed under localization. The same question has been posed for other classes of groups. For example, finite groups, perfect groups and torsion abelian groups are all not closed under localization. Nilpotent groups of class 2 are closed under localization.