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The Quarterly Journal of Mathematics Advance Access originally published online on October 19, 2006
The Quarterly Journal of Mathematics 2007 58(1):23-29; doi:10.1093/qmath/hal016
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© 2006. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

ELEMENTS WITH THE SAME NORMAL CLOSURE IN A METABELIAN GROUP

Gérard Endimioni{dagger}

C.M.I.–Université de Provence, 39, rue F. Joliot-Curie, F-13453 Marseille Cedex 13, France

{dagger} E-mail: endimion{at}gyptis.univ-mrs.fr

Received 17 February 2006;
  Abstract

It is known that in a free group, two elements a, b have the same normal closure if and only if the images of these elements have the same order in every finite quotient of the free group (Klyachko, 1999) or if and only if a is conjugate to b or b–1 (Magnus, 1931). Here, we prove that the result of Klyachko remains true in a finitely generated soluble group of derived length d ≤ 2. An example shows that the property fails when d = 3. Also, we give a counterpart of Magnus's result in the context of metabelian nilpotent group.


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