Resumen de Elements with the same normal closure in a metabelian group

Gérard Endimioni

• 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.