Ukrainian Mathematical Journal

, Volume 61, Issue 1, pp 1–13 | Cite as

Integral group ring of Rudvalis simple group

  • V. A. Bovdi
  • A. B. Konovalov

Using the Luthar–Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru . As a consequence, for this group we confirm the Kimmerle conjecture on prime graphs.


Conjugacy Class Nonnegative Integer Simple Group Group Ring Prime Graph 
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  1. 1.
    H. Zassenhaus, “On the torsion units of finite group rings,” in: Stud. Math. (in Honor of A. Almeida Costa), Instituto de Alta Cultura, Lisbon (1974), pp. 119–126.Google Scholar
  2. 2.
    I. S. Luthar and I. B. S. Passi, “Zassenhaus conjecture for A 5,” Proc. Indian Acad. Sci. Math. Sci., 99, No. 1, 1–5 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    I. S. Luthar and P. Trama, “Zassenhaus conjecture for S 5,” Commun. Algebra, 19, No. 8, 2353–2362 (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Hertweck, “Partial augmentations and Brauer character values of torsion units in group rings,” Commun. Algebra (to appear) (e-print: arXiv:math.RA/0612429v2).Google Scholar
  5. 5.
    V. Bovdi, C. Höfert, and W. Kimmerle, “On the first Zassenhaus conjecture for integral group rings,” Publ. Math. Debrecen., 65, No. 3-4, 291–303 (2004).zbMATHMathSciNetGoogle Scholar
  6. 6.
    V. Bovdi and A. Konovalov, “Integral group ring of the first Mathieu simple group,” Groups St Andrews, 1 (2005); London Math. Soc. Lect. Notes Ser., 339, 237–245 (2007).Google Scholar
  7. 7.
    M. Hertweck, “On the torsion units of some integral group rings,” Algebra Colloq., 13, No. 2, 329–348 (2006).zbMATHMathSciNetGoogle Scholar
  8. 8.
    M. Hertweck, “Torsion units in integral group rings of certain metabelian groups,” Proc. Edinburgh Math. Soc., 51, No. 2, 363–385 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    C. Höfert and W. Kimmerle, “On torsion units of integral group rings of groups of small order,” Lect. Notes Pure Appl. Math. Groups, Rings Group Rings, 248, 243–252 (2006).Google Scholar
  10. 10.
    V. A. Artamonov and A. A. Bovdi, “Integral group rings: groups of invertible elements and classical K-theory,” in: VINITI Series in Algebra, Topology, and Geometry, Vol. 27, VINITI, Moscow (1989), pp. 3–43, 232.Google Scholar
  11. 11.
    F. M. Bleher and W. Kimmerle, “On the structure of integral group rings of sporadic groups,” LMS J. Comput. Math. (Electronic), 3, 274–306 (2000).zbMATHMathSciNetGoogle Scholar
  12. 12.
    W. Kimmerle, “On the prime graph of the unit group of integral group rings of finite groups,” Contemp. Math. Groups, Rings and Algebras, 420, 215–228 (2006).MathSciNetGoogle Scholar
  13. 13.
    V. Bovdi and M. Hertweck, “Zassenhaus conjecture for central extensions of S 5,” J. Group Theory, 11, No. 1, 63–74 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    V. Bovdi, E. Jespers, and A. Konovalov, “Torsion units in integral group rings of Janko simple groups,” Preprint (2007) (e-print: arXiv:mathRA/0609435v1).Google Scholar
  15. 15.
    V. Bovdi and A. Konovalov, “Integral group ring of the Mathieu simple group M 24,” Preprint (2007) (e-print: arXiv:0705.1992vl).Google Scholar
  16. 16.
    V. Bovdi and A. Konovalov, “Integral group ring of the Mathieu simple group M 23,” Commun. Algebra, 36, 2670–2680 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    V. Bovdi, A. Konovalov, and S. Linton, “Torsion units in integral group ring of the Mathieu simple group M 22,” LMS J. Comput. Math., 11, 28–39 (2008).MathSciNetGoogle Scholar
  18. 18.
    V. Bovdi, A. Konovalov, and S. Siciliano, “Integral group ring of the Mathieu simple group M 12,” Rend. Circ. Mat. Palermo (2), 56, No. 1, 125–136 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    A. Rudvalis, “A rank 3 simple group of order 21433537 ⋅ 13 ⋅ 29. I,” J. Algebra, 86, No. 1, 181–218 (1984).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    S. D. Berman, “On the equation x m1 = 1 in an integral group ring,” Ukr. Mat. Zh., 7, 253–261 (1955).zbMATHGoogle Scholar
  21. 21.
    R. Sandling, “Graham Higman’s thesis ‘Units in group rings,”’ Lect. Notes Math. Integral Representations Appl. (Oberwolfach, 1980), 882, 93–116 (1981).MathSciNetGoogle Scholar
  22. 22.
    Z. Marciniak, J. Ritter, S. K. Sehgal, and A. Weiss, “Torsion units in integral group rings of some metabelian groups. II,” J. Number Theory, 25, No. 3, 340–352 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    J. A. Cohn and D. Livingstone, “On the structure of group algebras. I,” Can. J. Math., 17, 583–593 (1965).zbMATHMathSciNetGoogle Scholar
  24. 24.
    “The GAP Group,” GAP—Groups, Algorithms, and Programming, Version 4.4.10 ( (2007).
  25. 25.
    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups. With Computational Assistance from J. G. Thackray, Oxford University Press, Eynsham (1985).Google Scholar
  26. 26.
    C. Jansen, K. Lux, R. Parker, and R. Wilson, “An atlas of Brauer characters. Appendix 2 by T. Breuer and S. Norton,” London Math. Soc. Monogr. New Ser., 11 (1995).Google Scholar
  27. 27.
    V. Bovdi, A. Konovalov, R. Rossmanith, and C. Schneider, “LAGUNA—Lie AlGebras and UNits of group Algebras, Version 3.4, ( (2007).

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • V. A. Bovdi
    • 1
  • A. B. Konovalov
    • 2
  1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary
  2. 2.School of Computer ScienceUniversity of St AndrewsSt Andrews, ScotlandUK

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