Abstract
A group G is said to have the n-rewritable property Q(n) if for all elements g(1), g(2), ... , g(n) epsilon G, there exist two distinct permutations sigma, tau epsilon Sym(n) such that g(sigma(1))g(sigma(2)) ... g(sigma(n)) = g(tau(1))g(tau(2)) ... g(tau(n)). We show here that if G satisfies Q(n), then G has a characteristic subgroup N such that vertical bar G : N vertical bar and vertical bar N'vertical bar are both finite and have sizes bounded by functions of n. This extends the result of Blyth (1988) in [3] which asserts that if G satisfies Q(n) and if Delta is the finite conjugate center of the group, then vertical bar G : Delta vertical bar and vertical bar Delta'vertical bar are both finite with vertical bar G : Delta vertical bar bounded by a function of n. As a consequence, any group with Q(n) satisfies the permutational property P(m) with m bounded by a function of n. (C) 2011 Elsevier Inc. All rights reserved.