×

On the first Zassenhaus conjecture for integral group rings. (English) Zbl 1076.16028

Summary: It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed by I. S. Luthar and I. B. S. Passi [Proc. Indian. Acad. Sci., Math. Sci. 99, No. 1, 1-5 (1989; Zbl 0678.16008)] which sometimes permits an answer to this conjecture. We illustrate the method on certain explicit examples. We prove with additional arguments that the conjecture is valid for any 3-dimensional crystallographic point group. Finally we apply the method to generic character tables and establish a \(p\)-variation of the conjecture for the simple groups \(\text{PSL}(2,p)\).

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings

Citations:

Zbl 0678.16008

Software:

CHEVIE; LAGUNA
PDF BibTeX XML Cite
Full Text: arXiv