Remarks on Lifting Beauville Structures of Quasisimple Groups

  • Kay  MagaardEmail author
  • Christopher  Parker
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 123)


In previous work, we developed theorems which produce a multitude of hyperbolic triples for finite classical groups. We apply these theorems to prove a conjecture of Bauer, Catanese and Grunewald, which asserts that all non-abelian finite quasisimple groups except for the alternating group of degree five are Beauville groups. Here we show that our results can be used to show that certain split- and Frattini extensions of quasisimple groups are also Beauville groups. We also discuss some open problems for future investigations.

2000 Mathematics Subject Classification.

20E34 20F05 14J29 30F10 


  1. 1.
    M. Aschbacher, R.M. Guralnick, Some applications of the first cohomology group. J. Algebra 90(2), 446–460 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    B. Fairbairn, K. Magaard, C. Parker, Generation of finite quasisimple groups with an application to groups acting on Beauville surfaces. Proc. LMS appeared online 18 February, (2013)Google Scholar
  3. 3.
    W. Feit, On large Zsigmondy primes. Proc. Am. Math. Soc. 102(1), 29–36 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    R. Guralnick, G. Malle, Simple groups admit Beauville structures. J. Lond. Math. Soc. 85(3), 694–721 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    R.M. Guralnick, P.H. Tiep, Lifting in Frattini covers and a characterization of finite solvable groups. arXiv:1112.4559
  6. 6.
    G.A. Jones, Characteristically simple Beauville groups, I: Cartesian powers of alternating groups. arXiv:1304.5444
  7. 7.
    G.A. Jones, Characteristically simple Beauville groups, II: low rank and sporadic groups. arXiv:1304.5450
  8. 8.
    W. Jones, B. Parshall, On the 1-cohomology of finite groups of Lie type, in Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975)Google Scholar
  9. 9.
    Darren Semmen, The group theory behind modular towers. Séminaires Congrès 13, 343–366 (2006)MathSciNetGoogle Scholar
  10. 10.
    J.-P. Serre, Abelian \(l\)-adic representations and elliptic curves. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam (1968)Google Scholar
  11. 11.
    L.L. Scott, Matrices and cohomology, Ann. Math. (2) 105 (1977), no. 3, 473–492Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BirminghamEdgbastonUK

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