Archiv der Mathematik

, Volume 106, Issue 4, pp 305–314 | Cite as

Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples

Article

Abstract

For a subgroup L of the symmetric group \({S_{\ell}}\), we determine the minimal base size of \({GL_d(q) \wr L}\) acting on \({V_d(q)^{\ell}}\) as an imprimitive linear group. This is achieved by computing the number of orbits of GLd(q) on spanning m-tuples, which turns out to be the number of d-dimensional subspaces of Vm(q). We then use these results to prove that for certain families of subgroups L, the affine groups whose stabilisers are large subgroups of \({GL_{d}(q) \wr L}\) satisfy a conjecture of Pyber concerning bases.

Keywords

Permutation group Base size General linear group Imprimitive linear group Spanning sequence 

Mathematics Subject Classification

Primary 15A04 20B15 

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References

  1. 1.
    Bailey R. F., Cameron P. J.: Base size, metric dimension and other invariants of groups and graphs, Bull. London Math. Soc. 43, 209–242 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    C. Benbenishty, On actions of primitive groups, PhD thesis, Hebrew University, Jerusalem, 2005.Google Scholar
  3. 3.
    Burness T. C., Seress Á.: On Pyber’s base size conjecture, Trans. Amer. Math. Soc. 367, 5633–5651 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cameron P. J., Neumann P. M., Saxl J.: On groups with no regular orbits on the set of subsets, Arch. Math. 43, 295–296 (1984)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chan M.: The distinguishing number of the direct product and wreath product action, J. Algebr. Comb. 24, 331–345 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dolfi S.: Orbits of permutation groups on the power set, Arch. Math. 75, 321–327 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fawcett J. B.: The base size of a primitive diagonal group, J. Algebra 375, 302–321 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gluck D., Magaard K.: Base sizes and regular orbits for coprime affine permutation groups, J. London Math. Soc. 58, 603–618 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Liebeck M. W., Shalev A.: Simple groups, permutation groups, and probability, J. Amer. Math. Soc. 12, 497–520 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Liebeck M. W., Shalev A.: Bases of primitive linear groups, J. Algebra 252, 95–113 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Liebeck M. W., Shalev A.: Bases of primitive linear groups II, J. Algebra 403, 223–228 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Neumann P. M., Praeger C. E.: Cyclic matrices over finite fields, J. London Math. Soc. 52, 263–284 (1995)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Pyber L.: Asymptotic results for permutation groups, DIMACS Ser. Discrete Math. Theoret. Comp. Sci. 11, 197–219 (1993)MathSciNetMATHGoogle Scholar
  14. 14.
    Seress Á.: The minimal base size of primitive solvable permutation groups, J. London Math. Soc. 53, 243–255 (1996)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Seress Á.: Primitive groups with no regular orbits on the set of subsets, Bull. London Math. Soc. 29, 697–704 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Seress Á.: Permutation group algorithms, Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Centre for the Mathematics of Symmetry and Computation, School of Mathematics and StatisticsThe University of Western AustraliaCrawleyAustralia
  2. 2.King Abdulaziz UniversityJeddahSaudi Arabia

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