Abstract
Given a finite group \(G\) which possesses a non-abelian simple normal subgroup \(N\) having exactly four \(G\)-class sizes, we prove that \(N\) is isomorphic to PSL\((2, 2^a)\) with \(a\ge 2\). Thus, we obtain an extension for normal subgroups of the classic N. Itô’s theorem which characterizes those finite simple groups with exactly four class sizes.
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Akhlaghi, Z., Beltrán, A., Felipe, M.J., Khatami, M.: Structure of normal subgroups with three \(G\)-class sizes. Monatsh. Math. 167(1), 1–12 (2012)
Akhlaghi, Z., Beltrán, A., Felipe, M.J.: Normal sections, class sizes and solvability of finite groups. J. Algebra 399, 220–231 (2014)
Alemany, E., Beltrán, A., Felipe, M.J.: Nilpotency of normal subgroups having two \(G\)-class sizes. Proc. Amer. Math. Soc. 139, 2663–2669 (2011)
Beltrán, A.: Action with nilpotent fixed point subgroup. Arch. Math. (Basel) 69, 177–184 (1997)
Feit, W.: A characterization of the simple groups SL\((2,2^a)\). Amer. J. Math. 82, 281–300 (1969)
Fisman, E., Arad, Z.: A proof of Szep’s conjecture on non-simplicity of certain finite groups. J. Algebra 108, 340–354 (1987)
Glauberman, G.: Central elements in core-free groups. J. Algebra 4, 403–420 (1966)
Huppert, B.: Endliche Gruppen I. (German) Die Grundlehren der Mathematischen Wissenschaften. Band 134 Springer-Verlag, Berlin-New York (1967)
Itô, N.: On finite groups with given conjugate type. I. Nagoya Math. J. 6, 17–28 (1953)
Itô, N.: On finite groups with given conjugate types. II. Osaka J. Math. 7, 231–251 (1970)
Itô, N.: On finite groups with given conjugate types. III. Math. Z. 117, 267–271 (1970)
Isaacs, I.M.: Finite Group Theory. Graduate Studies in Mathematics 92. American Mathematical Society, Providence RI (2008)
Rebmann, J.: F-Gruppen. (German) Arch. Math. (Basel) 22, 225–230 (1971)
Acknowledgments
This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the first author is also supported by grant Fundació Bancaixa P11B2010-47.
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Communicated by J. S. Wilson.
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Beltrán, A., Felipe, M.J. Simplicity of normal subgroups and conjugacy class sizes. Monatsh Math 175, 485–490 (2014). https://doi.org/10.1007/s00605-013-0602-2
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DOI: https://doi.org/10.1007/s00605-013-0602-2