Bounded Engel group

Definition
A group $$G$$ is termed a $$n$$-Engel group if, for any $$x, y \in G$$, there exists $$n$$ such that:

$$[[\dots [x,y],y],\dots,y] = e$$

where the $$y$$ occurs $$n$$ times.

A group that is $$n$$-Engel for some positive integer $$n$$, is termed a bounded Engel group.

Weaker properties

 * Stronger than::Engel group
 * Stronger than::Nilpotent group: In fact a group of nilpotency class $$c$$ is a $$c$$-Engel group.