Lower central series (${\gamma }_i(G)$)

The lower central series of a group is defined by ${\gamma }_1(G)=G$ and ${\gamma }_{i+1}(G)=[{\gamma }_i(G),G]$. A group is nilpotent of class at most $c$ exactly when ${\gamma }_{c+1}(G)$ is trivial.

Let $G$ be a nilpotent group of class at most $s$ in which every element has order at most $r$. A subgroup generated by $k$ elements has order at mostIndeed, collection by the lower central series expresses every element as an ordered product of simple group commutators in the generators of weights at most $s$. There are at most $k+k^2+\cdots +k^s\le s{k}^s$ such commutators, and each exponent may be reduced to one of at most $r$ values.