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

Bovdi, Victor; Höfert, C.; Kimmerle, W.

Publ. Math. 65, No. 3-4, 291-303 (2004).

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

Keywords:

Zassenhaus conjecture; torsion units; integral group rings; finite groups; rational conjugation

Citations:

Zbl 0678.16008

Software:

CHEVIE; LAGUNA