2-Engel not implies class two for groups

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., 2-Engel group) need not satisfy the second group property (i.e., group of nilpotency class two)
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Statement

It is possible for a group to be a 2-Engel group but not a group of nilpotency class two.

Related facts

Opposite facts

• 2-Engel and 3-torsion-free implies class two for groups
• 2-Engel implies class three for groups

Proof

Example

Further information: Burnside group:B(3,3)

Note that any finite example must be a 3-group of nilpotency class three. The smallest example is a group of order $3^{7}=2187$. This is the Burnside group $B(3,3)$, defined as the quotient of free group:F3 by the relations that the cube of every element is the identity (this relation set is infinite, but we can get a finite presentation because the group does eventually turn out to be finit).