The Unit Of The Ring In The Cluster

Let G be a finite group,the main topic of this paper is the conjugate relationship between the torsion units of the integral group ring and the elements of G in rational group algebra QG. At the second part of the paper,some results about the torsion units of the integral group ring which is afforded by the direct product of some special groups are given,especially it is confirmed that the direct product of five degree alternative group A5and the cyclic group with order three C3satisfies Zassenhaus’conjecture. At the third part,using the so called Luthar-Passi method,it’s proved that the direct product of A5and the Dihedral group D6satisfies Zassenhaus’conjecture by compute the partial augmentations of the torsion untis of Z(A5×D6).Using the same method,the forth part will show that the symmetric group S6satisfies Kimmerle’s conjecture which is a weaker version of Zassenhaus’conjecture.