# Exponent three implies class three for groups

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## Contents

## Statement

Suppose is a group whose exponent is three. Then, is a group of nilpotency class three: it is a nilpotent group and its nilpotency class is at most three.

## Related facts

- Exponent two implies abelian, which follows from square map is endomorphism iff abelian

## Facts used

- Exponent three implies 2-Engel for groups (note that the analogous statement is
*not*true for Lie rings) - 2-Engel implies class three for groups (note that the analogous statement is true for Lie rings)

## Proof

The proof follows directly by combining Facts (1) and (2).