N-nilpotent group

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


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}.

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.