3-Engel Lie ring
This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
View a complete list of properties of Lie rings
VIEW RELATED: Lie ring property implications | Lie ring property non-implications |Lie ring metaproperty satisfactions | Lie ring metaproperty dissatisfactions | Lie ring property satisfactions | Lie ring property dissatisfactions
Definition
A Lie ring is termed a 3-Engel Lie ring if it satisfies the following:
![{\displaystyle [x,[x,[x,y]]]=0\ \forall \ x,y\in L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d81d310bdc24b3f67494241fd35078ba02cb536)
Facts
- 3-Engel and 2-torsion-free implies 2-local class three for Lie rings
- 3-Engel and (2,5)-torsion-free implies class six for Lie rings
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Lie ring of nilpotency class three | ||||
| Lie ring of 2-local nilpotency class three | ||||
| 2-Engel Lie ring |