What might be the condition for nontrivial homomorphism to exist between two groups?
Is it true that for $\psi: K \to H$, there is nontrivial homomorphic $\psi$ exists if they contain a common prime divisor $p$? Are there any established results on this? For example, I know $\psi: Z_3 \to S_3$ can be nontrivial by mapping the entire $Z_3$ to order 3 subgroup of $S_3$. But what about the general case.