Abstract
Let G be a group and ω(G) be the set of element orders of G. Let k ∈ ω(G) and m k (G) be the number of elements of order k in G. Let nse(G) = {m k (G): k ∈ ω(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 ≅ 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.
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Babai, A., Akhlaghi, Z. A new characterization of symmetric group by NSE. Czech Math J 67, 427–437 (2017). https://doi.org/10.21136/CMJ.2017.0700-15
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DOI: https://doi.org/10.21136/CMJ.2017.0700-15