TITLE: An example of a nilpotent group
QUESTION [7 upvotes]: Is there an example of a nilpotent group such that $G/G'$ is (non-trivial) torsion-free while $G$ is not? I cannot think of any example of this kind and I think that it is not proved any result like this, am I wrong?

REPLY [3 votes]: In fact, all relatively free groups of rank $k \geq 2$ in the variety defined by the law $[a_1, ... a_n]([x, y])^m$ for some fixed integers $n$ and $m$ satisfy this condition. Such groups are indeed nilpotent of degree $n$, all their commutators have finite orders dividing $m$ and their abelianization is the free abelian group of rank $k$.