Two-step nilpotent group

A group $G$ is two-step nilpotent whenEquivalently, its commutator subgroup is contained in the center of a group of $G$. This allows every word to be collected into powers of generators followed by powers of their pairwise commutators.

In a two-generated two-step nilpotent group, use $c=[x,y]=x^{-1}{y}^{-1}xy$. Centrality of $c$ gives $yx=xy{c}^{-1}$, so every word collects to $x^a{y}^b{c}^r$. The multiplication of these collected expressions is $(a,b,r)(a^{\prime },b^{\prime },r^{\prime })=(a+a^{\prime },b+b^{\prime },r+r^{\prime }-a^{\prime }b)$. Relations on the central commutator can reduce $r$ modulo its order. To turn collection into a word problem for a group algorithm, one must also prove the independence of the remaining coordinates, for example by constructing a group with that cocycle multiplication.