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:

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

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 $\mho ^{r}(G)$.