My question is : Is it possible to find two groups such that every function from one to other is a homomorphism ?

If not how can I prove it? .


1 Answer 1


If the second group is just the identity, it works.

Note: as @celtschk points out, that's the only example for the second group, as is easily shown. (See Greg Martin's remark below for proof)

  • 3
    $\begingroup$ The group with only the identity is known as the trivial group. Also, it should be worth mentioning that this is the only way to make it work. $\endgroup$
    – celtschk
    Sep 12, 2018 at 5:44
  • $\begingroup$ @celtschk Yes, that's the usual term. I wanted to be clear so I spelled it out. And I agree it's the only way. $\endgroup$
    – coffeemath
    Sep 12, 2018 at 5:45
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    $\begingroup$ Since it's an easy proof to write down: if there exists a non-identity element $a$ in the second group, then any function that sends the identity element of the first group to $a$ is not a homomorphism. $\endgroup$ Sep 12, 2018 at 6:51

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