## Zassenhaus conjecture for central extensions of $$S_5$$.(English)Zbl 1143.16032

A conjecture of Zassenhaus states that, for a finite group $$G$$, every torsion unit in its integral group ring $$\mathbb{Z}[G]$$ is conjugate to an element of $$\pm G$$ by a unit of the rational group ring $$\mathbb{Q}[G]$$. In this paper, the authors confirm this conjecture for the covering group $$\widetilde S_n=\langle g_1,\dots,g_{n-1},z\mid g_i^2=(g_jg_{j+1})^3=(g_kg_l)^2=z$$, $$z^2=[z,g_i]=1$$ for $$1\leq i\leq n-1$$, $$1\leq j\leq n-2$$, $$k\leq l-2\leq n-3\rangle$$ of the symmetric group $$S_5$$ and for the general linear group $$\text{GL}(2,5)$$. The result on $$\widetilde S_5$$, together with earlier known results about torsion units yields (Theorem 3) that if $$G$$ is a finite Frobenius group, then each torsion unit in $$\mathbb{Z}[G]$$ which is of prime-power order is conjugate to an element of $$\pm G$$ by a unit of $$\mathbb{Q}[G]$$.
The proofs are based on an extended version, developed by the second author [in Partial augmentations and Brauer character values of torsion units in group rings (Commun. Algebra, to appear), ArXiv eprint http://arxiv.org/abs/math.RA/0612429v2], of a procedure introduced by I. S. Luthar and the reviewer [in Proc. Indian Acad. Sci., Math. Sci. 99, No. 1, 1-5 (1989; Zbl 0678.16008)].

### 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 20C30 Representations of finite symmetric groups

Zbl 0678.16008

GAP; LAGUNA
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