Prove that the additive groups of $\Bbb Z$ and $\Bbb Q$ are not isomorphic.
It is hard to create a map that will show homomorphism.
Every homomorphism of $\mathbb Z$ into $\mathbb Q$ is determined by $f(1)$. It follows that there is no surjective homomorphism.
Prove that $\mathbb Q$ is not cyclic (try proof by contradiction) and show that being cyclic is invariant under isomorphisms.