Abstract
Let G be a group and \(\omega (G)=\{o(g)|g\in G\}\) be the set of element orders of G. Let \(k\in \omega (G)\) and \(s_k=|\{g\in G |o(g)=k\}|\). Let \(nse(G)=\{s_k|k\in \omega (G) \}\). In this paper, we prove that if G is a group and \(G_2 (4)\) is the Chevalley group such that \(nse(G)=nse(G_2 (4))\), then \(G\cong G_2 (4)\).
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References
Conway, J.H.; Curtis, R.T.; Norton, S.P.; Parker, R.A.; Wilson, R.A.: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, New York (1985)
Shi, W.: A new characterization of the sporadic simple groups. In: Proc. of the 1987 Singapore Conference on Group Theory, Walter de Gruyter, pp. 531–540 (1989).
Shao, C.; Shi, W.; Jiang, W.Q.: A new characterization of simple \(K_4\)-groups. Front. Math. China 3(3), 355–370 (2008)
Shen, R.; Shao, C.; Jiang, Q.; Shi, W.; Mazurov, V.: A new characterization of \(A5\). Monatsh. Math. 160, 337–341 (2010)
Herzog, M.: Finite simple groups divisible by only three primes. J. Algebra 10, 383–388 (1968)
Jafarzadeh, A.; Iranmanesh, A.: On simple \(K_n\)- groups for \(n=5, 6\). In: Trlifaj, J. (ed) Groups St. Andrews 2005 in Oxford, vol. 2. Cambridge University Press, pp. 517–526 (2007)
Khalili Asboei, A.R.S.S.S.; Amiri, A.; Iranmanesh, A.; Tehranian, A.: Characterization of sporadic simple groups by NSE and order. J. Algebra Appl. 12(2), 1250158 (2013)
Shao, C.; Jiang, Q.: A new characterization of some linear groups by nse. J. Algebra Appl. 13(2), 1350094 (2014)
Khatami, M.; Khosravi, B.; Akhlaghi, Z.: A new characterization for some linear groups. Monatsh Math. 163, 39–50 (2011)
Liu, S.: A characterization of \(L_3 (4)\). Sci. Asia 39, 436–439 (2013)
Liu, S.: NSE characterization of projective special group \(L_5 (2)\). Rend. Sem. Mat. Univ. Padova 132, 123–132 (2014)
Liu, S.: A characterization of projective special unitary group \(U_3 (5)\) by nse. Arab. J. Math. Sci. 20(1), 133–140 (2014)
Frobenius, G.: Verallgemeinerung des sylowschen satze. Berliner Sitz, pp. 981–993 (1895)
Miller, G.: Addition to a theorem due to Frobenius. Bull. Am. Math. Soc. 11, 6–7 (1904)
Hall, M.: The Theory of Groups. Macmilan, New York (1959)
Shi, W.J.: On simple \(K_4\)-group. Chin. Sci. Bull. 36(17), 1281–1283 (1991)
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Khangheshlaghi, M.J., Darafsheh, M.R. NSE characterization of the Chevalley group \(\varvec{G}_{\varvec{2}} {\varvec{(4)}}\) . Arab. J. Math. 7, 21–26 (2018). https://doi.org/10.1007/s40065-017-0182-4
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DOI: https://doi.org/10.1007/s40065-017-0182-4