The Magnus property for direct products

TL;DR

This paper investigates the Magnus property in groups, proving that it is preserved under direct products for certain residually finite-p groups, but also providing counterexamples with crystallographic groups.

Contribution

It establishes conditions under which the Magnus property is preserved in direct products and presents explicit counterexamples using crystallographic groups.

Findings

Direct product of residually finite-p groups with the Magnus property also has the property.

Counterexamples show the property does not always hold in direct products.

Crystallographic groups can serve as explicit examples of failure cases.

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.