Siberian Mathematical Journal

, Volume 52, Issue 1, pp 30–40 | Cite as

On finite groups isospectral to simple linear and unitary groups

  • A. V. Vasil’ev
  • M. A. Grechkoseeva
  • A. M. Staroletov
Article

Abstract

Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if LU 4(2), U 5(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.

Keywords

finite group spectrum of a group simple group linear group unitary group composition factor 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mazurov V. D., “Groups with prescribed spectrum,” Izv. Ural. Gos. Univ. Mat. Mekh., 7, No. 36, 119–138 (2005).MathSciNetGoogle Scholar
  2. 2.
    Grechkoseeva M. A., Shi W. J., and Vasilev A. V., “Recognition by spectrum for finite simple groups of Lie type,” Front. Math. China, 3, No. 2, 275–285 (2008).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Conway J. H., Curtis R. T., Norton S. P., Parker R. A., and Wilson R. A., Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press, Oxford (1985).MATHGoogle Scholar
  4. 4.
    Shi W. J., “A characteristic property of J1 and PSL2(2n),” Adv. Math. (in Chinese), 16, No. 4, 397–401 (1987).MATHGoogle 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).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Mazurov V. D., Xu M. C., and Cao H. P., “Recognition of the finite simple groups L3(2m) and U3(2m) from the orders of their elements,” Algebra and Logic, 39, No. 5, 324–334 (2000).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zavarnitsine A. V., “Recognition of the simple groups L3(q) by element orders,” J. Group Theory, 7, No. 1, 81–97 (2004).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Zavarnitsine A. V., “The weights of irreducible SL3(q)-modules in the defining characteristic,” Siberian Math. J., 45, No. 2, 261–268 (2004).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zavarnitsin A. V., “Recognition of simple groups U3(q) by element orders,” Algebra and Logic, 45, No. 2, 106–116 (2006).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Vasil’ev A. V., Grechkoseeva M. A., and Mazurov V. D., “On finite groups isospectral to simple symplectic and orthogonal groups,” Siberian Math. J., 50, No. 6, 965–981 (2009).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zsigmondy K., “Zür Theorie der Potenzreste,” Monatsh. Math. Phys., 3, 265–284 (1892).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Roitman M., “On Zsigmondy primes,” Proc. Amer. Math. Soc., 125, No. 7, 1913–1919 (1997).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Williams J. S., “Prime graph components of finite groups,” J. Algebra, 69, No. 2, 487–513 (1981).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Kondratiev A. S., “On prime graph components for finite simple groups,” Math. USSR-Sb., 67, No. 1, 235–247 (1990).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mazurov V. D., “Recognition of finite simple groups S4(q) by their element orders,” Algebra and Logic, 41, No. 2, 93–110 (2002).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kondrat’ev A. S. and Mazurov V. D., “Recognition of alternating groups of prime degree from their element orders,” Siberian Math. J., 41, No. 2, 294–302 (2000).MathSciNetCrossRefGoogle Scholar
  17. 17.
    Vasil’ev A. V., “On connection between the structure of a finite group and the properties of its prime graph,” Siberian Math. J., 46, No. 3, 396–404 (2005).MathSciNetCrossRefGoogle Scholar
  18. 18.
    Vasil’ev A. V. and Gorshkov I. B., “On recognition of finite simple groups with connected prime graph,” Siberian Math. J., 50, No. 2, 233–238 (2009).MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gorenstein D., Finite Groups, Harper and Row, New York (1968).MATHGoogle Scholar
  20. 20.
    Vasil’ev A. V. and Vdovin E. P., “An adjacency criterion for the prime graph of a finite simple group,” Algebra and Logic, 44, No. 6, 381–406 (2005).MathSciNetCrossRefGoogle Scholar
  21. 21.
    Vasil’ev A. V. and Grechkoseeva M. A., “Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2,” Algebra and Logic, 47, No. 5, 314–320 (2008).MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mazurov V. D. and Chen G. Y., “Recognizability of finite simple groups L4(2m) and U4(2m) by spectrum,” Algebra and Logic, 47, No. 1, 49–55 (2008).MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mazurov V. D., “Recognition of finite groups by a set of orders of their elements,” Algebra and Logic, 37, No. 6, 371–379 (1998).MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vasil’ev A. V. and Vdovin E. P., Cocliques of Maximal Size in the Prime Graph of a Finite Simple Group [Preprint, No. 225], Sobolev Institute of Mathematics, Novosibirsk (2009). See also http://arxiv.org/abs/0905.1164v1.Google Scholar
  25. 25.
    Testerman D. M., “A1-type overgroups of elements of order p in semisimple algebraic groups and the associated finite groups,” J. Algebra, 177, No. 1, 34–76 (1995).MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Zavarnitsine A. V., “Finite simple groups with narrow prime spectrum,” Sib. Electron. Math. Rep., 6, 1–12 (2009); http://semr.math.nsc.ru/v6/p1-12.pdf.MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  • A. V. Vasil’ev
    • 1
  • M. A. Grechkoseeva
    • 1
  • A. M. Staroletov
    • 1
  1. 1.Sobolev Institute of Mathematics and Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations