Rendiconti del Circolo Matematico di Palermo

, Volume 56, Issue 1, pp 125–136 | Cite as

Integral group ring of the Mathieu simple groupM 12

  • V. A. Bovdi
  • A. B. Konovalov
  • S. Siciliano


We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic groupM 12. As a consequence, we confirm for this group the Kimmerle’s conjecture on prime graphs.

1991 Mathematics Subject Classification

Primary 16S34 20C05 secondary 20D08 

Keys words and phrases

Zassenhaus conjecture Kimmerle conjecture torsion unit partial augmentation integral group ring 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Artamonov V. A., Bovdi A. A.,Integral group rings: groups of invertible elements and classical K-theory, In Algebra. Topology. Geometry, Vol. 27 (Russian), Itogi Nauki i Tekhniki, 3-43, 232. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989. Translated in J. Soviet Math.,57 (1991) no. 2, 2931–2958.Google Scholar
  2. [2]
    Berman S. D.,On the equation x m=1 in an integral group ring Ukrain. Mat. Ž.7 (1955), 253–261.zbMATHMathSciNetGoogle Scholar
  3. [3]
    Bleher F. M., Kimmerle W.,On the structure of integral group rings of sporadic groups, LMS J. Comput. Math.,3 (2000), 274–306 (electronic).zbMATHMathSciNetGoogle Scholar
  4. [4]
    Bovdi. V., Hertweck M.,Zassenhaus conjecture for central extensions of S 5 Preprint, 1–11, submitted, 2006.Google Scholar
  5. [5]
    Bovdi V. Höfert C., Kimmerle W.On the first Zassenhaus conjecture for integral group rings, Publ. Math. Debrecen,65 (2004), 291–303.zbMATHMathSciNetGoogle Scholar
  6. [6]
    Bovdi V., Jespers E., Konovalov A.,Porzion units in integral group rings of Janko simple groups, Preprint, 1–11, to appear, 2007.Google Scholar
  7. [7]
    Bovdi V., Konovalov A.,Integral group ring of the Mathieu simple group M 23, Comm. Algebra, 1–9, to appear, 2007.Google Scholar
  8. [8]
    Bovdi V. Konovalov A.,Integral group rings of the first Mathieu simple group, In Groups St. Andreus 2005, volume 1 of London Math. Soc. Lecture Note Ser. 237-245. Cambridge Univ. Press. Cambridge to appear, 2007.Google Scholar
  9. [9]
    Cohn J. A., Livingstone D.,On the structure of group algebras. I, Canad. J. Math.17 (1965), 583–593.zbMATHMathSciNetGoogle Scholar
  10. [10]
    The GAP Group.GAP —Groups, Algorithms, and Programming, Version 4.4 2006. (http: // Scholar
  11. [11]
    Gorenstein D.,The classification of finite simple groups. Vol. 1 The University Series in Mathematics, Plenum Press, New York, 1983.Google Scholar
  12. [12]
    Hertweck M.,On the torsion units of some integral group rings, Algebra Colloq.,13(2) (2006), 329–348.zbMATHMathSciNetGoogle Scholar
  13. [13]
    Hertweck M.,Partial augmentations and Brauer character values of torsion units in group rings, Preprint, submitted, 2005, 1–26.Google Scholar
  14. [14]
    Hertweck M.,Torsion units in integral group rings or certain metabelian groups, Proc. Edinb. Math. Soc., to appear, 2005, 1–22.Google Scholar
  15. [15]
    Höfert C. Kimmerle W.On torsion units of integral group rings of groups of small order, Groups, rings and group rings,248, of Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton FL, (2006), 243–252.Google Scholar
  16. [16]
    Kimmerle W.,On the prime graph of the unit group of integral group rings of finite groups, Groups rings and algebras. Papers in Honor of Donald S. Passman’s 65-th Birthday, Contemporary Mathematics, AMS, to appear, 2005.Google Scholar
  17. [17]
    Luthar I. S., Passi I. B. S.,Zassenhaus conjecture for A 5, Proc. Indian Acad. Sci. Math. Sci.,99(1) (1989), 1–5.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Luthar I. S., Trama P.,Zassenhaus conjecture for S 5, Comm. Algebra,19(8) (1991), 2353–2362.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Marciniak Z., Ritter J., Sehgal S. K., Weiss A.,Torsion units in integral group rings of some metabelian groups. II, J. Number Theory,25(3) (1987), 340–352.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • V. A. Bovdi
    • 1
    • 2
  • A. B. Konovalov
    • 3
  • S. Siciliano
    • 4
  1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary
  2. 2.Institute of Mathematics and InformaticsCollege of NyíregyházaNyíregyházaHungary
  3. 3.Department of MathematicsZaporozhye National UniversityZaporozhyeUkraine
  4. 4.Dipartimento di Matematica “E. De Giorgi”Universitá degli Studi di LecceLECCEItaly

Personalised recommendations