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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

  • Joanna B. Fawcett
  • Cheryl E. Praeger
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 GL d (q) on spanning m-tuples, which turns out to be the number of d-dimensional subspaces of V m (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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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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