# 3-Engel group

(Redirected from Group of Levi class two)
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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:

1. For all $x,y \in G$, we have that $[x,[x,[x,y]]]$ is the identity element of $G$ where $[\ , \ ]$ denotes the group commutator. In other words, the group is a bounded Engel group of Engel degree at most three.
2. Every subgroup arising as the normal subgroup generated by a singleton subset is a group of nilpotency class two.

### 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