3-Engel group
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Definition
A group is termed a 3-Engel group or group of Levi class two if it satisfies the following equivalent conditions:
- For all , we have that is the identity element of where denotes the group commutator. In other words, the group is a bounded Engel group of Engel degree at most three.
- Every subgroup arising as the normal subgroup generated by a singleton subset is a group of nilpotency class two.
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![{\displaystyle [\ ,\ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7717dae012f042ad66888075df87a1ebac4392a)
Equivalence of definitions
Further information: equivalence of definitions of 3-Engel group
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| group of nilpotency class two | ||||
| group of nilpotency class three |