3-Engel group

Definition
A group is termed a 3-Engel group or group of Levi class two if it satisfies the following equivalent conditions:


 * 1) For all $$x,y \in G$$, we have that $$[x,[x,[x,y]]]$$ is the identity element of $$G$$ where $$[\, \ ]$$ denotes the group commutator. In other words, the group is a defining ingredient::bounded Engel group of Engel degree at most three.
 * 2) Every subgroup arising as the normal subgroup generated by a singleton subset is a group of nilpotency class two.