Characterization of abelian-by-cyclic 3-rewritable groups

Abstract

Let n be an integer greater than 1. A group G is said to be n-rewritable (or a Qn-group) if for every n elements x1, x2,., xn in G there exist distinct permutations s and τ in Sn such that xσ(1)xσ(2)⋯xσ(n)=xτ(1)xτ(2)⋯xτ(n). In this paper we have completely characterized abelian-by-cyclic 3-rewritable groups: they turns out to have an abelian subgroup of index 2 or the size of derived subgroups is less than 6. In this paper, we also prove that G/F(G) is an abelian group of finite exponent dividing 12, where F(G) is the Fitting subgroup of G. © 2004, Universita di Padova. All rights reserved.