Groups with Many Rewritable Products

Abstract

For any integer n ≥ 2, a group G is said to have the n-rewritable property R _n if every infinite subset X of G contains n elements x ₁,…, x _n such that the product x ₁…x _n  = x _σ(1)…x _σ(n) for some permutation σ ≠ 1. We show here that if G satisfies R _n , then G has a subgroup N of finite index with a finite central subgroup A of N such that the exponent of (N/A)/Z(N/A) is finite and has size bounded by (n − 1)!. This extends the main result in [4] which asserts that a group G is an R _n group for some integer n if and only if G has a normal subgroup F such that G/F is finite, F is an FC-group, and the exponent of F/Z(F) is finite.

Key Words:

• FC-groups
• n-Permutable groups
• n-Rewritable groups
• R _n -Groups

2000 Mathematics Subject Classification:

• 20E25

ACKNOWLEDGMENT

This work was done during my visit to Prof. Donald S. Passman at the University of Wisconsin-Madison, USA. It was supported by the Egyptian Government through scientific channels between my university and his. I would like to thank him for his hospitality.

Dedicated to my professor Donald S. Passman on his retirement.

Notes

Communicated by S. Sehgal.