JOURNAL OF ALGEBRA 136, 86-91 (1991) On the Derived Length of Groups with Some Permutational Property MERCEDE MAJ* Dipartimento di Matematica e Applicazioni “R. Caccioppoli,” via Mezzocannone, 8, 80134 Napoli, Italy Communicated by Gernot Stroth Received January 23, 1989 1. INTRODUCTION Let G be a group and II be an integer z 2. We say that G is n-rewritable, or that G is in the class Q,, if and only if, for any (x1, x2, . . . . x,) E G”, there exist different permutations 0, z EC, such that %T(l)X,(Z) . . * X,(n) = X,(,)X,(,) . . . X,(n). A group G is said to be totally n-rewritable, or to be in the class P,, if and only if, for any (x1, x2, . . . . x,) E G”, there exists a non-trivial permutation 0 EC, such that x1x2 * - . XT? = X0(1)X,(2) . . . X,(n). Define P=Una2Pn and Q=Unr2Q,,. Obviously P,sQ,, so PEQ. The class P was first introduced in [lo], in the context of semigroups. Many authors have studied groups in P or in Q. It has been shown that P is the class of finite-by-abelian-by-finite groups (see [S]) and that this is also the class Q (see [2]). Hence P = Q. Obviously P, = Q2 is the class of abelian groups, while, for every n > 2, P, is always a proper subclass of Qn (see [2]). The class P, is the class of groups with derived subgroup of order at most 2 (see [4]), and groups in P4 are metabelian (see [7]), while groups in P5 or P, are soluble (see [3]). Also known is a complete descrip- tion of the class P, (see El, 681). In [2] R. D. Blyth and D. J. S. Robinson proved that a group in Q4 is always soluble and asked if a group in Q3 is metabelian. In this paper we give an affirmative answer to this question (see Section 2). Moreover we prove, more generally that the derived length I(G) * Work supported by M.P.I. 86 0021~8693/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector