Some Properties of Bell Groups

Abstract

For any integer n ≠ 0,1, a group G is said to be “n-Bell” if it satisfies the identity [x ⁿ ,y] = [x,y ⁿ ]. It is known that if G is an n-Bell group, then the factor group G/Z ₂(G) has finite exponent dividing 12n ⁵(n − 1)⁵. In this article we show that this bound can be improved. Moreover, we prove that every n-Bell group is n-nilpotent; consequently, using results of Baer on finite n-nilpotent groups, we give the structure of locally finite n-Bell groups. Finally, we are concerned with locally graded n-Bell groups for special values of n.

Key Words:

• n-Bell groups
• n-Levi groups
• n-Nilpotent groups

2000 Mathematics Subject Classification:

• 20F19
• 20E10
• 20F12

ACKNOWLEDGMENT

The author wishes to thank the referee for giving useful suggestions.

Notes

Communicated by M. R. Dixon.