TITLE: Kernel of a homomorphism from a free group into $\mathbb{Z}$.
QUESTION [9 upvotes]: Let $F$ be a non-abelian free group, that is, a free group of rank at least $2$, and let $\phi: F \rightarrow \mathbb{Z}$ be a nontrivial group homomorphism. How to prove that the kernel of $\phi$ is not finitely generated?

REPLY [4 votes]: For a completely different proof, that uses only the universal property of free groups, you could factor your map $\phi:F \to {\mathbb Z}$ as $\phi:F \to H \to {\mathbb Z}$, where $H$ is the wreath product ${\mathbb Z} \wr  {\mathbb Z}$, and observe that the kernel of $H \to {\mathbb Z}$ is infinitely generated.