# N-commutator

From Groupprops

## Definition

Suppose is a group, are (possibly equal, possibly distinct) elements of , and is an integer. The -commutator of in , denoted , is defined as:

Note that:

- For and , the -commutator is always the identity element.
- For , the -commutator is the inverse of a conjugate of the usual commutator , and for , the -commutator is a conjugate of the usual commutator.
- For any other , the -commutator is in the normal subgroup generated by , 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-derived subgroup is the subgroup generated by all the n-commutators. Again, this notion depends on the value of .
- n-derived series and n-lower central series
- n-nilpotent group
- n-solvable group