N-lower central series

Definition

Suppose $G$ is a group and $n$ is an integer. The $n$-lower central series of $G$ is defined as the following descending series $H_n, n \in \mathbb{N}$:

• The first member of the series is $G$, i.e., $H_1 = G$.
• For any $n$, $H_{n+1}$ is the subgroup of $G$ generated by all n-commutators $[x,y]_n = (xy)^ny^{-n}x^{-n}$ where $x \in H_n, y \in G$.