N-nilpotent group

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

Definition

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 $[x,y]_{n}:=(xy)^{n}y^{-n}x^{-n}$.

Terminology caution

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.