Hirsch induction and torsion units in group rings (Q1893234)

We study the torsion of the group of normalized integral units for polycyclic groups. We consider groups for which a \(p\)-local version of Hirsch induction can be applied. We prove that for such groups, as well as for all nilpotent and all metabelian groups, every finite group of units from \(Z \Gamma\) has an isomorphic copy inside \(\Gamma\). We also find new classes of groups with the unique trace property: each torsion unit has only one non vanishing Bass trace and hence may be a conjugate of a group element.