Torsional Units And Torsional Subgroups Of Some Finite Group Rings

Let be a finite group,the First Zassenhaus's conjecture is as follow if every torsion unit of ZG is conjugate to an element of in the unites of QG.In this paper we study the behavior of the first Zassenhaus's conjecture on the basis of previous studies.In this paper,we first collect some main facts about the finite group rings and extend some characteristics of the normalized torsion units.Then,considering the Hertweck-Luthar-Passi method,computing the partial augmentations of the torsion units of the direct product of the symmetric group S₅and the cyclic group C₃,we confirm the First Zassenhaus's conjecture about this group.Next,using the proposition of the finite group wreath product,we study the First Zassenhaus's conjecture about the wreath product of the cyclic group and the cyclic group.Finally,we prove two results of rational conjugation of the torsional subgroups of the integral group rings.