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:

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

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