Bell group

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

History

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

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 $n$.

Relation with other properties

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

References

• The structure of Bell groups by C. Delizia, R. R. Moghattam and A. M. Rhemtulla, appeared in Journal of Group Theory, 9, no. 1 (2006), 117-125
• On n-Levi groups by L. C. Kappe, appeared in Arch. Math. 47 (1986), 198 - 210

External links

• [http://www.math.klte.hu/publi/forthcoming/delizia_3457.pdf Locally graded Bell groups (PDF)