# 2-Engel not implies class two for groups

From Groupprops

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) neednotsatisfy the second group property (i.e., group of nilpotency class two)

View a complete list of group property non-implications | View a complete list of group property implications

Get more facts about 2-Engel group|Get more facts about group of nilpotency class two

## 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

## 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 . This is the Burnside group , 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).