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Strongly Real Beauville Groups

  • Ben FairbairnEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 123)

Abstract

A strongly real Beauville group is a Beauville group that defines a real Beauville surface. Here we discuss efforts to find examples of these groups, emphasising on the one extreme finite simple groups and on the other abelian and nilpotent groups. We will also discuss the case of characteristically simple groups and almost simple groups. En route we shall discuss several questions, open problems and conjectures as well as giving several new examples of infinite families of strongly real Beauville groups.

Keywords

Beauville structure Beauville group Beauville surface 

Notes

Acknowledgments

The author wishes to express his deepest gratitude to the organisers of the Beauville Surfaces and Groups 2012 conference held in the University of Newcastle without which this volume, and thus the opportunity to present these results here, would not have been possible. The author also wishes to thank Professor Gareth Jones for many invaluable comments on earlier drafts of this article, particularly regarding the results concerned with products of symmetric and alternating groups. Finally, the author wishes to thank the anonymous referee whose comments and suggestions have substantially improved the readability of this paper, particularly bearing in mind the wide breadth of the audience for this work.

References

  1. 1.
    M. Aschbacher, R. Guralnick, Some applications of the first cohomology group. J. Algebra 90(2), 446–460 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    N. Barker, N. Boston, N. Peyerimhoff, A. Vdovina, An infinite family of 2-groups with mixed Beauville structures, to appear in Int. Math. Res. Not. arXiv:1304.4480
  3. 3.
    N. Barker, N. Boston, N. Peyerimhoff, A. Vdovina, Regular algebraic surfaces isogenous to a higher product constructed from group representations using projective planes, preprint (2011). arXiv:1109.6053
  4. 4.
    N.W. Barker, N. Boston, B.T. Fairbairn, A note on Beauville \(p\)-groups. Exp. Math. 21(3), 298–306 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    N. Barker, N. Boston, N. Peyerimhoff, A. Vdovina, New examples of Beauville surfaces. Monatsh. Math. 166(3–4), 319–327 (2012). doi: 10.1007/s00605-011-0284-6 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    I. Bauer, F. Catanese, F. Grunewald, Beauville surfaces without real structures, Geometric Methods in Algebra and Number Theory, vol. 235, Progress in Mathematics (Birkhuser, Boston, 2005)CrossRefGoogle Scholar
  7. 7.
    I. Bauer, F. Catanese, F. Grunewald, Chebycheff and Belyi polynomials, Dessins d‘Enfants, Beauville surfaces and group theory. Mediterr. J. Math. 3, 121–146 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    H.U. Besche, B. Eick, E.A. O’Brien, The groups of order at most \(2{\,}000\). Electron. Res. Announc. Am. Math. Soc. 7, 1–4 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    H.U. Besche, B. Eick, E.A. O’Brien, A millennium project: constructing small groups. Int. J. Algebra Comput. 12(5), 623–644 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    N. Boston, A survey of Beauville \(p\)-groups, in Proceedings of Conference on Beauville Surfaces and Group, ed. by I. Bauer, S. Garion, A. Vdovina (Newcastle, 2012)Google Scholar
  12. 12.
    N. Boston, M.R. Bush, F. Hajir, Heuristics for \(p\)-class towers of imaginary quadratic fields, preprint (2011). arXiv:1111.4679v1
  13. 13.
    N. Boston, Embedding 2-groups in groups generated by involutions. J. Algebra 300(1), 73–76 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    F. Catanese, Fibered surfaces, varieties isogenous to a product and related moduli spaces. Am. J. Math. 122(1), 1–44 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    F. Catanese, Moduli spaces of surfaces and real structures. Ann. Math. 158(2), 577–592 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, An ATLAS of Finite Groups (Clarendon Press, Oxford, 1985)Google Scholar
  17. 17.
    J.D. Dixon, B. Mortimer, Permutation Groups (Springer, New York, 1996)CrossRefzbMATHGoogle Scholar
  18. 18.
    B.T. Fairbairn, Some exceptional Beauville structures. J. Group Theory 15(5), 631–639 (2012). arXiv:1007.5050
  19. 19.
    B.T. Fairbairn, K. Magaard, C.W. Parker, Corrigendum to Generation of finite simple groups with an application to groups acting on Beauville surfaces, to appear in the Proc. Lond. Math. SocGoogle Scholar
  20. 20.
    B.T. Fairbairn, K. Magaard, C.W. Parker, Generation of finite simple groups with an application to groups acting on Beauville surfaces. Proc. Lond. Math. Soc. 107(5), 1220 (2013). doi: 10.1112/plms/pdt037
  21. 21.
    Y. Fuertes, G. González-Diez, On Beauville structures on the groups \(S_n\) and \(A_n\). Math. Z. 264(4), 959–968 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Y. Fuertes, G.A. Jones, Beauville surfaces and finite groups. J. Algebra 340, 13–27 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    S. Garion, On Beauville structures for \(PSL(2, q)\), preprint (2010). arXiv:1003.2792
  24. 24.
    S. Garion, M. Larsen, A. Lubotzky, Beauville surfaces and finite simple groups. J. Reine Angew. Math. 666, 225–243 (2012)zbMATHMathSciNetGoogle Scholar
  25. 25.
    R. Guralnick, G. Malle, Simple groups admit Beauville structures. J. Lond. Math. Soc. (2) 85(3), 694–721 (2012)Google Scholar
  26. 26.
    P. Hall, The Eulerian functions of a group. Q. J. Math. 7, 134–151 (1936)CrossRefGoogle Scholar
  27. 27.
    G.A. Jones, Beauville surfaces and groups: a survey, to appear Fields Inst. CommunGoogle Scholar
  28. 28.
    G.A. Jones, Characteristically simple Beauville groups I: Cartesian powers of alternating groups. preprint (2013). arXiv:1304.5444
  29. 29.
    G.A. Jones, Characteristically simple Beauville groups II: low rank and sporadic groups. preprint (2013). arXiv:1304.5450
  30. 30.
    G.A. Jones, Primitive permutation groups containing a cycle, preprint (2012). arXiv:1209.5169v1
  31. 31.
    R. Steinberg, Generators for simple groups. Can. J. Math. 14, 277–283 (1962)CrossRefzbMATHGoogle Scholar
  32. 32.
    The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.5.7; 2012. http://www.gap-system.org
  33. 33.
    R.A. Wilson et al., Atlas of finite group representations, version 3. http://brauer.maths.qmul.ac.uk/Atlas/ (2005) onwards
  34. 34.
    R.A. Wilson, Standard generators for sporadic simple groups. J. Algebra 184(2), 505–515 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    R.A. Wilson, The Finite Simple Groups, vol. 251, Graduate Texts in Mathematics (Springer, London, 2009)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Economics, Mathematics and Statistics, BirkbeckUniversity of LondonLondonUK

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