The Magnus property for direct products

Abstract

A group G is said to have the Magnus property if the following holds: whenever two elements x,y have the same normal closure, then x is conjugate to y or its inverse. We prove: Let p be an odd prime, and let G,H be residually finite-p groups with the Magnus property. Then the direct product of G and H has the Magnus property. By considering suitable crystallographic groups, we give an explicit example of finitely generated, torsion-free, residually-finite groups G,H with the Magnus property such that the direct product of G and H does not have the Magnus property.