Skip to main content
SpringerLink
Log in

On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q)

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

The spectrum of a finite group is the set of its element orders. Two groups are isospectral whenever they have the same spectra. We consider the classes of finite groups isospectral to the simple symplectic and orthogonal groups B 3(q), C 3(q), and D 4(q). We prove that in the case of even characteristic and q > 2 these groups can be reconstructed from their spectra up to isomorphisms. In the case of odd characteristic we obtain a restriction on the composition structure of groups of this class.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mazurov V. D., “Groups with prescribed spectrum,” Izv. Ural. Gos. Univ. Mat. Mekh., 7, No. 36, 119–138 (2005).

    MathSciNet  Google Scholar 

  2. Mazurov V. D., “Characterization of finite groups by the sets of element orders,” Algebra and Logic, 36, No. 1, 23–32 (1997).

    Article  MathSciNet  Google Scholar 

  3. Mazurov V. D., “Recognition of finite simple groups S 4(q) by their element orders,” Algebra and Logic, 41, No. 2, 93–110 (2002).

    Article  MathSciNet  Google Scholar 

  4. Gorshkov I. B., “Recognition by spectrum for finite simple groups with orders having prime divisors at most 17,” Siberian Electronic Math. Reports, 7, 14–20 (2010).

    MathSciNet  Google Scholar 

  5. 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).

  6. Zsigmondy K., “Zür Theorie der Potenzreste,” Monatsh. Math. Phys., 3, 265–284 (1892).

    Article  MathSciNet  MATH  Google Scholar 

  7. Roitman M., “On Zsigmondy primes,” Proc. Amer. Math. Soc., 125, No. 7, 1913–1919 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 ().

  9. 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).

    Article  MathSciNet  Google Scholar 

  10. Mazurov V. D., “Characterization of finite groups by sets of element orders,” Algebra and Logic, 36, No. 1, 23–32 (1997).

    Article  MathSciNet  Google Scholar 

  11. Vasil’ev A. V., “Recognition of the groups S4(q) from their element orders,” Algebra i Logika, 41, No. 2, 130–142 (2002).

    MathSciNet  MATH  Google Scholar 

  12. Buturlakin A. A., “Spectra of finite symplectic and orthogonal groups,” Siberian Adv. in Math., 21, No. 3, 176–210 (2011).

    Article  Google Scholar 

  13. 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).

    Article  MathSciNet  Google Scholar 

  14. 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).

    Article  MathSciNet  Google Scholar 

  15. Buturlakin A. A., “Spectra of finite linear and unitary groups,” Algebra and Logic, 47, No. 2, 91–99 (2008).

    Article  MathSciNet  Google Scholar 

  16. Stensholt E., “Certain embeddings among finite groups of Lie type,” J. Algebra, 53, No. 1, 136–187 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  17. Deng H. W. and Shi W. J., “Recognition of some finite simple groups of type D n(q) by spectrum,” Int. J. Algebra Comput., 19, No. 5, 681–698 (2009).

    Article  Google Scholar 

  18. 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., 13, No. 1, 145–154 (2007).

    Article  MathSciNet  Google Scholar 

  19. Gorenstein D., Lyons R., and Solomon R., The Classification of the Finite Simple Groups, Amer. Math. Soc., Providence (1998). (Math. Surveys and Monogr.; V. 40, No. 3).

  20. Carter R. W., Simple Groups of Lie Type, John Wiley and Sons, London (1972).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. M. Staroletov

Authors
  1. A. M. Staroletov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Staroletov.

Additional information

Original Russian Text Copyright © 2012 Staroletov A. M.

The author was supported the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh-3669.2010.1 and MK-2136.2010.1), the Program “Development of the Scientific Potential of Higher Education” of the Russian Federal Agency for Education (Grant 2.1.1.10726), the Federal Target Program “Scientific and Educational Personnel of Innovation Russia” for 2009–2013 (State Contracts 02.740.11.0429 and 02.740.11.5191), and the Lavrent’ev Grant for Young Scientists of the Siberian Division of the Russian Academy of Sciences (Resolution No. 43 of 04.02.2010).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 3, pp. 663–671, May–June, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Staroletov, A.M. On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q). Sib Math J 53, 532–538 (2012). https://doi.org/10.1134/S0037446612020334

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446612020334

Keywords

Search

Navigation