Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Groups isospectral to the degree 10 alternating group

Abstract

The spectrum of a finite group is the set of its element orders. We describe the composition structure of every finite group with the same spectrum as that of the alternating group of degree 10 and not isomorphic to it. This group is isomorphic to the semidirect product of the abelian {3, 7}-group, which contains an element of order 21, by the symmetric group of degree 5.

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

References

  1. 1.

    Lucido M. S. and Moghaddamfar A. R., “Groups with complete prime graph connected components,” J. Group Theory, 7, No. 3, 373–384 (2004).

  2. 2.

    Mazurov V. D., “Recognition of finite groups by the set of their element orders,” Algebra and Logic, 37, No. 6, 371–379 (1998).

  3. 3.

    Staroletov A. M., “Insolubility of the finite groups that are isospectral to the alternating group of degree 10,” Sib. Elektron. Mat. Izv., 5, 20–24 (2008).

  4. 4.

    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).

  5. 5.

    Gorenstein D., Finite Groups, Harper and Row, New York (1968).

  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).

  7. 7.

    Zavarnitsin A. V., “Recognition of alternating groups of degrees r +1 and r +2 for prime r and the group of degree 16 by their element order sets,” Algebra and Logic, 39, No. 6, 370–377 (2000).

  8. 8.

    Mazurov V. D., “On the set of the element orders of a finite group,” Algebra and Logic, 33, No. 1, 49–55 (1994).

  9. 9.

    Mazurov V. D. and Khukhro E. I., “On groups admitting a group of automorphisms whose centralizer has bounded rank,” Sib. Elektron. Mat. Izv., 3, 257–283 (2006).

  10. 10.

    GAP—Groups, Algorithms and Programming, version 4.4.9 (http://www.gap-system.org).

  11. 11.

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

  12. 12.

    Holt D. F. and Plesken W., Perfect Groups, Clarendon Press, Oxford (1989).

Download references

Author information

Correspondence to A. M. Staroletov.

Additional information

Original Russian Text Copyright © 2010 Staroletov A. M.

The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-344.2008.1), the Russian Federal Agency for Education (Grant 2.1.1.419), and the Lavrent’ev Young Scientists Competition of the Russian Academy of Sciences (Resolution No. 43 of 04.02.2010).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 3, pp. 638–648, May–June, 2010.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Staroletov, A.M. Groups isospectral to the degree 10 alternating group. Sib Math J 51, 507–514 (2010). https://doi.org/10.1007/s11202-010-0053-0

Download citation

Keywords

  • spectrum of a group
  • isospectral groups
  • recognition of groups by spectrum
  • simple group
  • Frobenius group
  • alternating group