Exponent three implies class three for groups

Statement

Suppose $G$ is a group whose exponent is three. Then, $G$ 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

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