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

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.

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Correspondence to Joanna B. Fawcett.

Additional information

This research forms part of the Discovery Project Grant DP130100106 of the second author, funded by the Australian Research Council. The first author is supported by this Grant.

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Fawcett, J.B., Praeger, C.E. Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples. Arch. Math. 106, 305–314 (2016). https://doi.org/10.1007/s00013-016-0890-6

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Mathematics Subject Classification

  • Primary 15A04
  • 20B15

Keywords

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