# Engel group

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

## Definition

A group $G$ is termed an Engel group or nil group or nilgroup, if, given any two elements $x,y \in G$, there exists a $n$ such that the iterated commutator: $[[ \dots [x,y],y],y],\dots],y] = e$

where $e$ denotes the identity element, and $y$ occurs $n$ times.

If there exists a $n$ that works for all pairs of elements of $G$, then we say that $G$ is a $n$-Engel group. A $n$-Engel group, for some $n$, is termed a bounded Engel group. Note that sometimes the term Engel group is used for bounded Engel group.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
locally nilpotent group every finitely generated subgroup is nilpotent 2-locally nilpotent group|FULL LIST, MORE INFO