# N-nilpotent group

Suppose $G$ is a group and $n$ and $c$ are integers, with $c$ nonnegative. We say that $G$ is $n$-nilpotent with class at most $c$ if the $(c+1)^{th}$ member of the n-lower central series of $G$ is the trivial subgroup. The n-lower central series is a series starting with the first member $G$, where each member is defined as the subgroup generated by all n-commutators between elements of the previous member and all of $G$. Here, the $n$-commutator of $x,y \in G$ is defined as $[x,y]_n := (xy)^ny^{-n}x^{-n}$.
Note that this differs from the notion of p-nilpotent group. Thus, when somebody uses $n$-nilpotent for a specific numerical value of $n$ that is prime, it is more likely that they are referring to the concept of p-nilpotent than the concept of n-nilpotent mentioned here.