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

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
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