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

摘要

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 Luthara€“Passi method for checking the Zassenhaus conjecture for specific groups e??o. In this note, we report that the Luthara€“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 a€“ the non-split covering group of e?‘?4, and one of order 72 of isomorphism type (e??? ?— e???) ?— e??·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 $mathbb{Z}G$.