# 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 is a group and and are integers, with nonnegative. We say that is -nilpotent with class at most if the member of the n-lower central series of is the trivial subgroup. The n-lower central series is a series starting with the first member , where each member is defined as the subgroup generated by all n-commutators between elements of the previous member and all of . Here, the -commutator of is defined as .

## Terminology caution

Note that this differs from the notion of p-nilpotent group. Thus, when somebody uses -nilpotent for a specific numerical value of 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.