An n-rewritability criterion for infinite groups

Abstract

Let n > 1 be an integer. A group G is said to be n-rewritable, whenever for any subset {x1, . . ., xn} of elements of G, there exist distinct permutations τ, σ of the set {1, 2, . . ., n} such that xτ(1) · · · xτ(n) = xσ(1) · · · xσ(n). In this paper we show that an infinite group G is n-rewritable if and only if for every n infinite subsets X1, . . ., Xn of G there exist distinct permutations τ, σ of the set {1, 2, . . ., n} such that Xτ(1) · · · Xτ(n) ∩ Xσ(n) · · · Xσ(n) ≠ 0.