# 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

## Definition

A group is termed an **Engel group** or **nil group** or **nilgroup**, if, given any two elements , there exists a such that the iterated commutator:

where denotes the identity element, and occurs times.

If there exists a that works for all pairs of elements of , then we say that is a -Engel group. A -Engel group, for some , 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 | ||

nilpotent group | 2-locally nilpotent group, Locally nilpotent group|FULL LIST, MORE INFO | |||

bounded Engel group | |FULL LIST, MORE INFO | |||

2-locally nilpotent group | subgroup generated by two elements is always nilpotent | |FULL LIST, MORE INFO |