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    This chapter has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Bächle, Andreas and Margolis, Leo 2018. HeLP: a GAP package for torsion units in integral group rings. Journal of Software for Algebra and Geometry, Vol. 8, Issue. 1, p. 1.

    Bächle, Andreas and Margolis, Leo 2017. On the prime graph question for integral group rings of 4-primary groups I. International Journal of Algebra and Computation, Vol. 27, Issue. 06, p. 731.

    Gildea, Joe 2016. Torsion units for some almost simple groups. Czechoslovak Mathematical Journal, Vol. 66, Issue. 2, p. 561.

    Gildea, Joe and Tylyshchak, Alexander 2016. Torsion units in the integral group ring of PSL(3, 4). Journal of Algebra and Its Applications, Vol. 15, Issue. 01, p. 1650013.

    GILDEA, JOE 2013. ZASSENHAUS CONJECTURE FOR INTEGRAL GROUP RING OF SIMPLE LINEAR GROUPS. Journal of Algebra and Its Applications, Vol. 12, Issue. 06, p. 1350016.

    BOVDI, VICTOR and KONOVALOV, ALEXANDER 2012. INTEGRAL GROUP RING OF THE MATHIEU SIMPLE GROUP M24. Journal of Algebra and Its Applications, Vol. 11, Issue. 01, p. 1250016.

    BOVDI, V. A. KONOVALOV, A. B. and LINTON, S. 2011. TORSION UNITS IN INTEGRAL GROUP RINGS OF CONWAY SIMPLE GROUPS. International Journal of Algebra and Computation, Vol. 21, Issue. 04, p. 615.

    Bovdi, Victor and Konovalov, Alexander 2010. Torsion units in integral group ring of Higman-Sims simple group. Studia Scientiarum Mathematicarum Hungarica, Vol. 47, Issue. 1, p. 1.

    Bovdi, V. A. and Konovalov, A. B. 2008. Integral Group Ring of the Mathieu Simple GroupM23. Communications in Algebra, Vol. 36, Issue. 7, p. 2670.

    Bovdi, V. A. Konovalov, A. B. and Linton, S. 2008. Torsion Units in Integral Group Ring of the Mathieu Simple Group M22. LMS Journal of Computation and Mathematics, Vol. 11, Issue. , p. 28.

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  • Print publication year: 2007
  • Online publication date: May 2010

Integral group ring of the first Mathieu simple group

    • By Victor Bovdi, Institute of Mathematics, University of Debrecen, P.O. Box 12, H–4010 Debrecen; Institute of Mathematics and Informatics, College of Nyíregyháza, Sóstói út 31/b, H–4410 Nyíregyháaza, Hungary, Alexander Konovalov, Department of Mathematics, Zaporozhye National University, 66 Zhukovskogo str., 69063, Zaporozhye, Ukraine; current address: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, B–1050 Brussel, Belgium
  • Edited by C. M. Campbell, University of St Andrews, Scotland, M. R. Quick, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
  • Publisher: Cambridge University Press
  • https://doi.org/10.1017/CBO9780511721212.016
  • pp 237-245
Summary

Abstract

We investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the simple Mathieu group M11. As a consequence, for this group we confirm the conjecture by Kimmerle about prime graphs.

Introduction and main results

Let V (ℤG) be the normalized unit group of the integral group ring ℤG of a finite group G. The following famous conjecture was formulated by H. Zassenhaus in [15]:

Conjecture 1 (ZC) Every torsion unit u ∈ V(ℤG) is conjugate within the rational group algebra ℚG to an element of G.

This conjecture is already confirmed for several classes of groups but, in general, the problem remains open, and a counterexample is not known.

Various methods have been developed to deal with this conjecture. One of the original ones was suggested by I. S. Luthar and I. B. S. Passi [12, 13], and it was improved further by M. Hertweck [9]. Using this method, the conjecture was proved for several new classes of groups, in particular for S5 and for some finite simple groups (see [4, 9, 10, 12, 13]).

The Zassenhaus conjecture appeared to be very hard, and several weakened variations of it were formulated (see, for example, [3]). One of the most interesting modifications was suggested by W. Kimmerle [11]. Let us briefly introduce it now.

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Groups St Andrews 2005
  • Online ISBN: 9780511721212
  • Book DOI: https://doi.org/10.1017/CBO9780511721212
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