# 4-Engel implies locally nilpotent for groups

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., 4-Engel group) must also satisfy the second group property (i.e., locally nilpotent group)

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## Statement

The statement has the following equivalent formulations:

- Any 4-Engel group is a locally nilpotent group.
- Any finitely generated 4-Engel group is a nilpotent group. (In fact, we can work out an explicit bound on the nilpotency class of the group in terms of the size of the generating set).

## Related facts

### Similar facts for groups

- 2-Engel implies class three for groups
- 2-Engel and 3-torsion-free implies class two for groups
- 3-Engel implies locally nilpotent for groups
- 3-Engel and (2,5)-torsion-free implies class four for groups

### Similar facts for Lie rings

- Zelmanov's theorem on Engel Lie rings says that all Engel Lie rings are locally nilpotent
- 2-Engel implies class three for Lie rings
- 2-Engel and 3-torsion-free implies class two for Lie rings
- 3-Engel and (2,5)-torsion-free implies class six for Lie rings