Characterization of 3-Rewritable Finite Nilpotent Groups

Let n be an integer greater than 1. A group G is said to be “ n-rewritable” whenever for every n elements x ₁,…,x _n of G, there exist distinct permutations τ, σ on the set {1,2,…, n} such that x _τ(1) ··· x _τ(n) = x _{σ (1)} ··· x _{σ (n)}. In this article, we complete the classification of 3-rewritable finite nilpotent groups and prove that a finite nilpotent group G is 3-rewritable if and only if G has an abelian subgroup of index 2 or the derived subgroup has order < 6.