Kernel of epimorphism from $F_{k}$ to $Z_{n}^k$

Question

How can be proved what all the epimorphisms from free group $F_{k}$ to $Z_{n}^k$ have the same kernel ?
I guess where something must be done with the fact what $F_{k}$ is isomorphic to free product of $k$ $Z$.

Answer 1

Hint: the kernel of every such epimorphism is the verbal subgroup of $F_k$ determined by the set of words $\{x^n, [x,y]\}$.

Answer 2

Let $D$ denotes the derived subgroup of $F_{k}$ and $\varphi:F_k\twoheadrightarrow F_k/D\simeq\mathbb Z_n^k$ the canonical projection. Then any epimorphism $f:F_k\twoheadrightarrow\mathbb Z_n^k$ can be factored through $\varphi$ and we get $f^{*}:F_k/D\simeq\mathbb Z_n^k\twoheadrightarrow\mathbb Z_n^k$. By Nakayama's Lemma, $f^{*}$ is an isomorphism.