Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 427-437, 2017

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi

Received December 26, 2015.   First published March 20, 2017.

Azam Babai, Department of Mathematics, University of Qom, Alghadir Blvd., Qom, P.O. Box 37185-3766, Iran, e-mail:; Zeinab Akhlaghi, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, District 6, Hafez Avenue No. 424, 15914 Tehran, Iran, e-mail:

Abstract: Let $G$ be a group and $\omega(G)$ be the set of element orders of $G$. Let $k\in\omega(G)$ and $m_k(G)$ be the number of elements of order $k$ in $G$. Let nse$(G) = \{m_k(G) k \in\omega(G)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$(G)=$ nse$(S_r)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ 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.

Keywords: set of the numbers of elements of the same order; prime graph

Classification (MSC 2010): 20D06, 20D15

DOI: 10.21136/CMJ.2017.0700-15

Full text available as PDF.

  [1] N. Ahanjideh, B. Asadian: NSE characterization of some alternating groups. J. Algebra Appl. 14 (2015), Article ID 1550012, 14 pages. DOI 10.1142/S0219498815500127 | MR 3270051 | Zbl 1320.20016
  [2] A. K. Asboei: A new characterization of PGL$(2,p)$. J. Algebra Appl. 12 (2013), Article ID 1350040, 5 pages. DOI 10.1142/S0219498813500400 | MR 3063479 | Zbl 1278.20013
  [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. MR 3005112 | Zbl 1261.20025
  [4] G. Frobenius: Verallgemeinerung des Sylow'schen Satzes. Berl. Ber. (1895), 981-993. (In German.) DOI 10.3931/e-rara-18880 | JFM 26.0158.01
  [5] D. Gorenstein: Finite Groups. Harper's Series in Modern Mathematics, Harper and Row, Publishers, New York (1968). MR 0231903 | Zbl 0185.05701
  [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. DOI 10.1112/plms/s3-31.2.149 | MR 0374247 | Zbl 0313.20004
  [7] M. Hall, Jr.: The Theory of Groups. The Macmillan Company, New York (1959). MR 0103215 | Zbl 0084.02202
  [8] B. Huppert: Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen 134, Springer, Berlin (1967). (In German.) DOI 10.1007/978-3-642-64981-3 | MR 0224703 | Zbl 0217.07201
  [9] M. Khatami, B. Khosravi, Z. Akhlaghi: A new characterization for some linear groups. Monatsh. Math. 163 (2011), 39-50. DOI 10.1007/s00605-009-0168-1 | MR 2787581 | Zbl 1216.20022
  [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.) DOI 10.1007/BF02674599 | MR 1762188 | Zbl 0956.20007
  [11] C. Shao, Q. Jiang: A new characterization of some linear groups by nse. J. Algebra Appl. 13 (2014), Article ID 1350094, 9 pages. DOI 10.1142/S0219498813500941 | MR 3119655 | Zbl 1286.20021
  [12] W. J. Shi: A new characterization of the sporadic simple groups. Group Theory Proc. Conf., Singapore, 1987, Walter de Gruyter, Berlin (1989), 531-540. DOI 10.1515/9783110848397-040 | MR 0981868 | Zbl 0657.20017
  [13] L. Weisner: On the Sylow subgroups of the symmetric and alternating groups. Am. J. Math. 47 (1925), 121-124. DOI 10.2307/2370639 | MR 1506549 | JFM 51.0117.02

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]