Say I have two nontrivial groups, $G_1,G_2$. Is there always a nontrivial homomorphism $f : G_1 \to G_2$ ?

My first instinct was no, and I figured I could easily find a counterexample. I tried $G_1 = \mathbb{Z_3}, G_2=\mathbb{Z_2}$ and some other examples, but always there is at least one nontrivial homomorphism.

Is the answer to my question yes after all? How do you prove it?

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    $\begingroup$ Your guess of $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ is a good one. Why do you think there is a non-trivial homomorphism between these groups? $\endgroup$ – carmichael561 Oct 20 '17 at 16:30
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    $\begingroup$ @carmichael561 My non-trivial homomoprhism is $0 \to0, 1\to 1,2 \to 1$. Am I missing something glaringly obvious here? $\endgroup$ – ghthorpe Oct 20 '17 at 16:37
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    $\begingroup$ Call this map $\phi$. You must have $\phi(x+y)=\phi(x)+\phi(y)$ for all $x$ and $y$, but in this case $\phi(2)=1\neq 0=\phi(1)+\phi(1)$. $\endgroup$ – carmichael561 Oct 20 '17 at 16:41
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    $\begingroup$ To show that there is no non-trivial homomorphism $\phi:\mathbb{Z}/3\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$, note that any homomorphism satisfies $\phi(1)=0$, since $0=\phi(1+1+1)=\phi(1)+\phi(1)+\phi(1)$. $\endgroup$ – carmichael561 Oct 20 '17 at 16:46
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    $\begingroup$ I appreciate your intuition, because it's right. Easiest thing ever if you just look at the kernel. $\endgroup$ – Randall Oct 20 '17 at 18:33

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