3-Engel group

From Groupprops
Jump to: navigation, search
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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions


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