Archiv der Mathematik

, Volume 106, Issue 4, pp 305–314

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


DOI: 10.1007/s00013-016-0890-6

Cite this article as:
Fawcett, J.B. & Praeger, C.E. Arch. Math. (2016) 106: 305. doi:10.1007/s00013-016-0890-6


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.


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

Mathematics Subject Classification

Primary 15A04 20B15 

Funding information

Funder NameGrant NumberFunding Note
Australian Research Council
  • DP130100106

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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