Suppose that we have a group $G$ generated by a set $S$ of elements with the only family of relations being that all commutators are central. Equivalently, $G$ is the largest group of nilpotency class $2$ generated by $S$. Now suppose that we have two equivalent words $x\_1x\_2\cdots x\_n$ and $y\_1y\_2\cdots y\_n$, where each $x\_i$ and $y\_j$ is a generator. Does it follow that $x\_i = y\_i$ for all $i$?


Intuitively, the answer should be "no", since that's the usual answer to these kinds of questions, but I am having trouble constructing a counterexample.



Analyzing words in a "free" group of nilpotency class 2
In the free class 2 nilpotent group with generators $x,y$, we have, for example, $xy^2x=yx^2y$.


More generally, since nilpotent groups have polynomial growth, they cannot contain free semigroups of rank greater than 1, so such equations must exist for all nilpotent groups.


