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

@article{Fawcett2016BaseSO,
  title={Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples},
  author={Joanna B. Fawcett and Cheryl E. Praeger},
  journal={Archiv der Mathematik},
  year={2016},
  volume={106},
  pages={305-314}
}
For a subgroup L of the symmetric group $${S_{\ell}}$$Sℓ, we determine the minimal base size of $${GL_d(q) \wr L}$$GLd(q)≀L acting on $${V_d(q)^{\ell}}$$Vd(q)ℓ 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… 
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Bases of twisted wreath products

A proof of Pyber's base size conjecture

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