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A new characterization of symmetric group by NSE. (English) Zbl 1458.20022

Summary: 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.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D06 Simple groups: alternating groups and groups of Lie type
20B30 Symmetric groups
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