Engel group

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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