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Siberian Mathematical Journal

, Volume 49, Issue 2, pp 246–256 | Cite as

Properties of element orders in covers for Ln(q) and Un(q)

  • A. V. ZavarnitsineEmail author
Article

Abstract

We show that if a finite simple group G, isomorphic to PSLn(q) or PSUn(q) where either n ≠ 4 or q is prime or even, acts on a vector space over a field of the defining characteristic of G; then the corresponding semidirect product contains an element whose order is distinct from every element order of G. We infer that the group PSLn(q), n ≠ 4 or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka Notebook.

Keywords

modular representation weight element order recognition 

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References

  1. 1.
    The Kourovka Notebook: Unsolved Problems in Group Theory, 14th ed., Sobolev Inst. Mat., Novosibirsk (1999).Google Scholar
  2. 2.
    Zavarnitsine A. V., “The weights of irreducible SL3(q)-modules in the defining characteristic,” Siberian Math. J., 45, No. 2, 261–268 (2004).CrossRefMathSciNetGoogle Scholar
  3. 3.
    Vasil’ev A. V. and Grechkoseeva M. A., “On recognition by spectrum of finite simple linear groups over fields of characteristic 2,” Siberian Math. J., 46, No. 4, 593–600 (2005).CrossRefMathSciNetGoogle Scholar
  4. 4.
    Mazurov V. D. and Zavarnitsine A. V., “On element orders in coverings of the simple groups Ln(q) and Un(q),” Proc. Steklov Inst. Math., Suppl. 1, 145–154 (2007).Google Scholar
  5. 5.
    Brandl R. and Shi W., “The characterization of PSL2(q) by its element orders,” J. Algebra, 163, No. 1, 109–114 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Mazurov V. D. and Zavarnitsine A. V., “Element orders in coverings of symmetric and alternating groups,” Algebra and Logic, 38, No. 3, 159–170 (1999).CrossRefMathSciNetGoogle Scholar
  7. 7.
    Testerman D. M., “A 1-Type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups,” J. Algebra, 177, No. 1, 34–76 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Carter R. W., “Centralizers of semisimple elements in the finite classical group,” Proc. London Math. Soc. (3), 42, No. 1, 1–41 (1981).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hanson D., “On a theorem of Sylvester and Schur,” Canad. Math. Bull., 16, 195–199 (1973).zbMATHMathSciNetGoogle Scholar
  10. 10.
    Rohrbach H. and Weis J., “Zum finiten Fall des Bertrandschen Postulates,” J. Reine Angew. Math., Bd 214/5, 432–440 (1964/65).MathSciNetGoogle Scholar
  11. 11.
    Mazurov V. D., “On the set of orders of elements of a finite group,” Algebra and Logic, 33, No. 1, 49–55 (1994).CrossRefMathSciNetGoogle Scholar
  12. 12.
    Zavarnitsine A. V., “Recognition of the simple groups L 3(q) by element orders,” J. Group Theory, 7, No. 1, 81–97 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Jantzen J. C., Representations of Algebraic Groups. Second edition, Amer. Math. Soc., Providence, RI (2003) (Math. Surveys Monogr.; 107).zbMATHGoogle Scholar
  14. 14.
    Suprunenko I. D. and Zalesskii A. E., “Fixed vectors for elements in modules for algebraic groups,” Internat. J. Algebra Comput., 17, No. 5–6, 1249–1261 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bourbaki N., Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées. Paris: Hermann, 1975 (Actualités Sci. Indust.; No. 1364).Google Scholar
  16. 16.
    Steinberg R., Lectures on Chevalley Groups, Yale University, New Haven (1968).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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