N-derived subgroup

Definition
Suppose $$G$$ is a group and $$n$$ is an integer. The $$n$$-derived subgroup of $$G$$ is the subgroup of $$G$$ generated by the set of all n-commutators of elements of $$G$$. Here, the $$n$$-commutator of two elements $$x,y \in G$$ (possibly equal, possibly distinct) is defined as:

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

Note that the $$n$$-derived subgroup depends on the value of $$n$$.

The $$n$$-derived subgroup of $$G$$ may be denoted $$[G,G]_n$$.

Facts

 * For all $$n$$, the $$n$$-derived subgroup is contained in the usual derived subgroup.
 * For the cases $$n = 2, n = -1$$, it is exactly equal to the derived subgroup. This follows from the facts that square map is endomorphism iff abelian and inverse map is automorphism iff abelian.
 * The $$n$$-derived subgroup is also contained in the subgroup generated by all $$n^{th}$$ powers. In the case that the group $$G$$ is a p-group and $$n = p^r$$, the subgroup generated by $$n^{th}$$ powers is the same as the agemo subgroup $$\mho^r(G)$$.