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

Joanna B. Fawcett; Cheryl E. Praeger

doi:10.1007/s00013-016-0890-6

paper

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

arXiv:1602.04913 · doi:10.1007/s00013-016-0890-6

Abstract

For a subgroup $L$ of the symmetric group $S_{ℓ}$ , we determine the minimal base size of $G L_{d} (q) ≀ L$ acting on $V_{d} (q)^{ℓ}$ as an imprimitive linear group. This is achieved by computing the number of orbits of $G L_{d} (q)$ 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 $G L_{d} (q) ≀ L$ satisfy a conjecture of Pyber concerning bases.

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The base size of a primitive diagonal group

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

Abstract

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