I'm currently stuck on this problem:
Let $m$ and $n$ be positive integers, and $F_k$ be the free group generated by $k$ elements. Prove that there is a homomorphism from $F_n$ onto $F_m$ if and only if $m$ is less than or equal to $n$.
My thought was to try abelianizing the free groups and using a homomorphism from the abelianized free groups to $\mathbb{Z}^n$ and $\mathbb{Z}^m$ respectively, but that has so far proved unfruitful.
Can someone help me find a better way to approach this problem?