Revisiting the Zassenhaus conjecture on torsion units for the integral group rings of small groups

Abstract

In recent years several new restrictions on integral partial augmentations for torsion units of $\mathbb {Z} G$ have been introduced, which have improved the effectiveness of the Luthar–Passi method for checking the Zassenhaus conjecture for specific groups G. In this note, we report that the Luthar–Passi method with the new restrictions are sufficient to verify the Zassenhaus conjecture with a computer for all groups of order less than 96, except for one group of order 48 – the non-split covering group of S ₄, and one of order 72 of isomorphism type (C ₃ × C ₃) ⋊ D ₈. To verify the Zassenhaus conjecture for this group we give a new construction of normalized torsion units of $\mathbb {Q} G$ that are not conjugate to elements of $\mathbb {Z} G$.