Groups with Many Rewritable Products

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 Curzio , M. , Longobardi , P. , Maj , M. , Rhemtulla , A. ( 1992 ). Groups with many rewritable products . Proc. AMS. 115 ( 4 ): 931 – 934 .[Crossref], [Web of Science ®] , [Google Scholar]] 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.