Czechoslovak Mathematical Journal, online first, 11 pp.

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.

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