# N-abelian iff (1-n)-abelian

From Groupprops

## Statement

Suppose is an integer and is a group. Then, is a -abelian group (see n-abelian group) if and only if is a -abelian group.

## Related facts

## Proof

The idea is to show that the condition for being -abelian on is equivalent to the condition for being -abelian on . Since the inverse map is bijective, varying over all of also varies over all of .

**Given**: A group , elements such that .

**To prove**: .

**Proof**: This is straightforward group element manipulation.

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