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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

Citations:

Zbl 0678.16008

Software:

GAP; LAGUNA
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References:

[1] Berman S. D., Ukrain. Mat. Z. 7 pp 253– (1955)
[2] Bessenrodt C., Sém. Lothar. Combin. 33 pp B33a– (1994)
[3] DOI: 10.2140/pjm.1999.187.215 · Zbl 0930.20012
[4] DOI: 10.1016/0022-4049(94)00091-V · Zbl 0834.20005
[5] Bovdi V., Publ. Math. Debrecen 65 pp 291– (2004)
[6] Cli{\currency} G. H., Canad. J. Math. 32 pp 596– (1980) · Zbl 0439.16011
[7] Cohn J. A., I. Canad. J. Math. 17 pp 583– (1965) · Zbl 0132.27404
[8] Dokuchaev M. A., Canad. J. Math. 48 pp 1170– (1996) · Zbl 0870.16020
[9] DOI: 10.1080/00927879708825991 · Zbl 0881.16020
[10] Farkas D. R., Canad. Math. Bull. 43 pp 60– (2000) · Zbl 0949.16032
[11] Hertweck M., Algebra Colloq. 13 pp 329– (2006)
[12] DOI: 10.1515/jgth.2000.022 · Zbl 0959.16019
[13] DOI: 10.1017/S1446788700034881
[14] DOI: 10.1007/BF02874643 · Zbl 0678.16008
[15] DOI: 10.1080/00927879108824263 · Zbl 0729.16021
[16] DOI: 10.1016/0022-314X(87)90037-0 · Zbl 0611.16007
[17] Morris A. O., Proc. Roy. Soc. Edinburgh Sect. A 108 pp 145– (1988)
[18] Schur I., J. Reine Angew. Math. 139 pp 155– (1911) · JFM 42.0154.02
[19] Zassenhaus H., Lisbon pp 119– (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.