Define the rank of a group $G$ as the minimal size of a subset $S$ such that $S$ generates $G$. I want to prove that, given $G$ and $H$ two groups, if there exists a surjective homomorphism $f$ between $G$ and $H$ then rank$(G)$ $\geq$ rank$(H)$.
I thought of this: let $S$ be a minimal generating set of $G$. Let us consider $f(S)$. Then clearly $\vert S \vert \geq \vert f(S) \vert$. It would then suffice to prove that $f(S)$ generates $H$. However, I see no good reason for this... Does anyone see a way to prove it?