For "$n=1$" the answer is "yes."  A group is abelian iff its generators commute.

Let $G_0=G$ be a group and let it be generated by $X_0=X$.  For each $n>0$ let $G_n=[G_{n-1},G_{n-1}]$ and let $X_n=[X_{n-1},X_{n-1}]$.  If $X_n$ is trivial, is $G_n$ trivial?