N-commutator

Definition
Suppose $$G$$ is a group, $$x,y$$ are (possibly equal, possibly distinct) elements of $$G$$, and $$n$$ is an integer. The $$n$$-commutator of $$x,y$$ in $$G$$, denoted $$[x,y]_n$$, is defined as:

$$[x,y]_n = (xy)^ny^{-n}x^{-n}$$

Note that:


 * For $$n = 0$$ and $$n = 1$$, the $$n$$-commutator is always the identity element.
 * For $$n = -1$$, the $$n$$-commutator is the inverse of a conjugate of the usual commutator $$[x,y]$$, and for $$n = 2$$, the $$n$$-commutator is a conjugate of the usual commutator.
 * For any other $$n$$, the $$n$$-commutator is in the normal subgroup generated by $$[x,y]$$, but it may generate a strictly smaller normal subgroup.

Related notions

 * n-abelian group is a group where the n-commutator of any two elements is the identity element. Note that the notion of n-abelian depends on the value of $$n$$.
 * n-derived subgroup is the subgroup generated by all the n-commutators. Again, this notion depends on the value of $$n$$.
 * n-derived series and n-lower central series
 * n-nilpotent group
 * n-solvable group