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2-Coverings for exceptional and sporadic simple groups

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In this paper we prove that if G is a finite exceptional simple group of Lie type, then G admits a 2-covering if, and only if, it is one of the following groups: \({G_2(2^{a}), F_4(3^{a}), G_2(2)', ^{2}G_2(3)', ^{2}F_4(2)'}\). Furthermore, if G is a finite sporadic simple group, then G admits a 2-covering if, and only if, G = M11.

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References

  1. D. Bubboloni, Coverings of the Symmetric and Alternating Groups, http://arxiv.org/abs/1009.3866.

  2. Bubboloni D., Lucido M. S.: Coverings of linear groups. Comm. Algebra, 30, 2143–2159 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bubboloni D., Lucido M. S., Weigel Th.: Generic 2-coverings of finite groups of Lie type. Rend. Sem. Mat. Univ. Padova 115, 209–252 (2006)

    MathSciNet  MATH  Google Scholar 

  4. D. Bubboloni, M. S. Lucido, and Th. Weigel, 2-coverings of classical groups, http://arxiv.org/abs/1102.0660.

  5. J. H. Conway et al. Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, Oxford University Press, Eynsham, 1985.

  6. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.5.4 of 16-Jun-2012, (http://www.gap-system.org).

  7. Guralnick R. M., Malle G.: Products of conjugacy classes and fixed point spaces. J. Amer. Math. Soc. 25, 77–121 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guralnick R. M., Malle G.: Simple groups admit Beauville structures. J. London Math. Soc. 85, 694–721 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kleidman P. B.: The maximal subgroups of the Steinberg triality groups \({^3{D}_4(q)}\) and of their automorphism groups. J. Algebra 115, 182–199 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. M. W. Liebeck, J. Saxl, and G. M. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc. (3) 65 (1992), 297–325.

    Google Scholar 

  11. M. S. Lucido, On the n-covers of exceptional groups of Lie type, Groups St. Andrews 2005. Vol. 2, 621–623, London Math. Soc. Lecture Note Ser., 340, Cambridge Univ. Press, Cambridge, 2007.

  12. Norton S. P., Wilson R. A.: The maximal subgroups of F 4(2) and its automorphism group. Comm. Algebra 17, 2809–2824 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Weigel T.: Generation of exceptional groups of Lie-type. Geom. Dedicata 41, 63–87 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. R. A. Wilson, The finite simple groups, Graduate Texts in Mathematics, 251. Springer-Verlag London, Ltd., London, 2009.

  15. Zsigmondy K.: Zur Theorie der Potenzreste. Monatsh. für Math. u. Phys. 3, 265–284 (1892)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Marco Antonio Pellegrini.

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Pellegrini, M.A. 2-Coverings for exceptional and sporadic simple groups. Arch. Math. 101, 201–206 (2013). https://doi.org/10.1007/s00013-013-0562-8

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