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

Bovdi, Victor; Hertweck, Martin

J. Group Theory 11, No. 1, 63-74 (2008).

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)].

Reviewer: Inder Bir Singh Passi (Chandigarh)

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

Keywords:

integral group rings; Zassenhaus conjecture; partial augmentations; Brauer characters; torsion units in group rings; finite Frobenius groups; covering groups; symmetric groups

Citations:

Zbl 0678.16008

Software:

GAP; LAGUNA

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