# A problem on group homomorphism .

Show that if $$G_1$$ and $$G_2$$ are two finite groups and the number of distinct group homomorphisms from $$G_1$$ to $$H$$ is equal to that of $$G_2$$ to $$H$$ for every finite group $$H$$, then that $$G_1$$ and $$G_2$$ are isomorphic .

It's a problem in IMS 2014 .

Actually, I have tried to show $$|G_1|= |G_2|$$ , primarily, and if there exists an onto homomorphism from $$G_1$$ onto $$G_2$$, then our assertion will be done. But, I can't approach properly! Here, I have found an answer on Aops

But, I want a different approach .

• What machinery do you have at hand? If you've read Artin's book or Dummit's, how many chapters have you read? Jul 27, 2019 at 16:44
• @Mikhail D, Sir, I have read upto group homomorphisms, group actions, and some portions of finite abelian groups . Jul 27, 2019 at 16:56
• After some contemplatiin, I arrived too at what boils down to the proof you reference - now I can't unthink it :( -- Why do you desire a different approach? Is there something you on't like with this one? May 17, 2020 at 14:59
• @Hagen von Eitzen,Sir, actually, I thought the proof may use isomorphism theorems, or some theorems from Sylow's, but it mainly uses number theory, so I thought if it can be done on another way. May 18, 2020 at 4:12