Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 427–437

A new characterization of symmetric group by NSE



Let G be a group and ω(G) be the set of element orders of G. Let kω(G) and mk(G) be the number of elements of order k in G. Let nse(G) = {mk(G): kω(G)}. Assume r is a prime number and let G be a group such that nse(G) = nse(Sr), where Sr is the symmetric group of degree r. In this paper we prove that GSr, if r divides the order of G and r2 does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.


set of the numbers of elements of the same order prime graph 

MSC 2010

20D06 20D15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Ahanjideh, B. Asadian: NSE characterization of some alternating groups. J. Algebra Appl. 14 (2015), Article ID 1550012, 14 pages.Google Scholar
  2. [2]
    A. K. Asboei: A new characterization of PGL(2, p). J. Algebra Appl. 12 (2013), Article ID 1350040, 5 pages.Google Scholar
  3. [3]
    A. K. Asboei, S. S. S. Amiri, A. Iranmanesh, A. Tehranian: A characterization of symmetric group S r, where r is prime number. Ann. Math. Inform. 40 (2012), 13–23.MathSciNetMATHGoogle Scholar
  4. [4]
    G. Frobenius: Verallgemeinerung des Sylow’schen Satzes. Berl. Ber. (1895), 981–993. (In German.) doiGoogle Scholar
  5. [5]
    D. Gorenstein: Finite Groups. Harper’s Series in Modern Mathematics, Harper and Row, Publishers, New York, 1968.Google Scholar
  6. [6]
    K. W. Gruenberg, K. W. Roggenkamp: Decomposition of the augmentation ideal and of the relation modules of a finite group. Proc. Lond. Math. Soc., III. Ser. 31 (1975), 149–166.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    M. Hall, Jr.: The Theory of Groups, The Macmillan Company, New York, 1959.MATHGoogle Scholar
  8. [8]
    B. Huppert: Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen 134, Springer, Berlin, 1967. (In German.)Google Scholar
  9. [9]
    M. Khatami, B. Khosravi, Z. Akhlaghi: A new characterization for some linear groups. Monatsh. Math. 163 (2011), 39–50.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    A. S. Kondrat’ev, V. D. Mazurov: Recognition of alternating groups of prime degree from their element orders. Sib. Math. J. 41 (2000), 294–302; translation from Sib. Mat. Zh. 41 (2000), 359–369. (In Russian.)CrossRefMATHGoogle Scholar
  11. [11]
    C. Shao, Q. Jiang: A new characterization of some linear groups by nse. J. Algebra Appl. 13 (2014), Article ID 1350094, 9 pages.Google Scholar
  12. [12]
    W. J. Shi: A new characterization of the sporadic simple groups. Group Theory. Proc. Conf., Singapore, 1987, Walter de Gruyter, Berlin, 1989, pp. 531–540.Google Scholar
  13. [13]
    L. Weisner: On the Sylow subgroups of the symmetric and alternating groups. Am. J. Math. 47 (1925), 121–124.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of QomQomIran
  2. 2.Faculty of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

Personalised recommendations