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 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
|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|