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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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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DOI: https://doi.org/10.1007/s00013-016-0890-6