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.