Engel group

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 $$y$$ 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.