Bell group

Origin
The term $$n$$-Bell group was introduced by Kappe in the paper On n-Levi groups by Kappe. The general term Bell group was introduced by Delizia, Moghattam and Rhemtulla in their paper The structure of Bell groups.

Definition with symbols
A group $$G$$ is termed an $$n$$-Bell group if for any $$x, y \in G$$, we have: $$[x,y^n] = [x^n,y]$$ where the square braces denote the commutator.

A group $$G$$ is termed a Bell group if it is $$n$$-Bell for some positive integer $$n$$.

Equivalent properties

 * Kappe group: There is a notion of $$n$$-Kappe group. While it is not true that every $$n$$-Bell group is $$n$$-Kappe, it is true that every Bell group is a Kappe group and vice versa.

Stronger properties

 * Abelian group
 * Bounded-exponent group
 * 2-Engel group