On the torsion units of the integral group ring of finite projective special linear groups

ABSTRACT

H. J. Zassenhaus conjectured that any unit of finite-order and augmentation one in the integral group ring of a finite group G is conjugate in the rational group algebra to an element of G. One way to verify this is showing that such unit has the same distribution of partial augmentations as an element of G and the HeLP Method provides a tool to do that in some cases. In this paper, we use the HeLP Method to describe the partial augmentations of a hypothetical counterexample to the conjecture for the projective special linear groups.

KEYWORDS:

• Group rings
• torsion units
• Zassenhaus conjecture

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 16U60, 16S34
• Secondary 20C05

Acknowledgment

We are very thankful to Andreas Bächle and Leo Margolis for several useful conversations on Zassenhaus conjecture.