Say, there are two groups $A$ and $B$. We are given that $\mathrm{Hom}(A,G)$ and $\mathrm{Hom}(B,G)$ are isomorphic, where $G$ is another group that may or may not be trivial. What can we say about the relationship between $A$ and $B$? Not really a homework problem, just wondering.
Asked
Active
Viewed 54 times
6
-
2If $G$ is trivial, not much. – Asinomás Jul 09 '15 at 22:06
-
1what do you mean that the two hom sets are the same? If $A\ne B$, then the two hom sets are never the same. – Ittay Weiss Jul 09 '15 at 22:06
-
Oh yeah, we don't really have a sensible way of defining this do we? – Asinomás Jul 09 '15 at 22:07
-
1oh, sure we do, and it's actually a very interesting question. One can ask for the hom sets to be isomorphic as sets, or as groups. One can ask for the isomorphism to hold for more than just a single group $G$. One can ask for naturality conditions. – Ittay Weiss Jul 09 '15 at 22:10
-
1Yes, I meant that the Hom groups are isomorphic. – Walmart Jul 09 '15 at 22:11
-
Isomorphic with respect to what algebraic structure on the hom-sets? Note that the pointwise group structure on $\hom(A, G)$ does not depend on the group structure of $A$. – Rob Arthan Jul 09 '15 at 22:18
-
I just meant that the group of homomorphisms is isomorphic. – Walmart Jul 09 '15 at 22:22
-
1How are you making the set of homomorphisms into a group? – Rob Arthan Jul 09 '15 at 22:23
1 Answers
4
If $G$ is just a single fixed group, then one can say very little about $A$ and $B$. For instance, if $G$ is a one-element group, then the hom groups are always isomorphic. Requiring the hom groups to be isomorphic for a larger class of groups may salvage more information. For instance, it is not hard to show (though a bit tricky) that for finite groups $A,B$ if the hom sets (not hom groups even, just the sets) are isomorphic for all finite groups $G$, then the two groups must be isomorphic. This fails for infinite groups, though the same result is recovered if one requires the hom sets to be naturally isomorphic for all groups $G$. This result is just a special case of an elementary result in category theory about representability.
Ittay Weiss
- 79,840
- 7
- 141
- 236
-
-
I suggest Leinster's "Basic Category Theory" for the basics of CT. – Ittay Weiss Jul 09 '15 at 22:32
-
I wasn't asking for a category theory primer. Just saying "naturally isomorphic for all groups $G$" does not make it clear what categories, what functors and what natural equivalences you have in mind and is unlikely to help the OP. – Rob Arthan Jul 09 '15 at 22:38
-
You are right Rob. But this is standard CT stuff and if OP is interested, they have a book reference. I agree with you the answer would have been better if I elaborated, but I don't have the time to do that now. Feel free to edit if you wish. – Ittay Weiss Jul 09 '15 at 22:45