Describe all non-injective group homomorphisms from $\mathbb Z$ to $\mathbb Q^*$.
So $\mathbb Z=\langle 1\rangle$. So when $f(z) \neq 0, z \neq 0$ we will get a non-injective homomorphism.
What's the way of thinking about this when we speak of infinite groups? I usually solve these questions using the fact that the order an element's image divides the element's order, but $1$ has infinite order. What do now?