## 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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