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?