# 3-Engel group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## 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.

### 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 |