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.