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

TL;DR

This paper determines the minimal base sizes of certain imprimitive linear groups and explores orbit counts of $GL_d(q)$ on spanning tuples, providing evidence for a conjecture on bases in affine groups.

Contribution

It introduces a method to compute orbit counts of $GL_d(q)$ on spanning tuples, linking them to subspace counts, and applies this to verify a conjecture of Pyber for specific affine groups.

Findings

Number of $GL_d(q)$ orbits on spanning $m$-tuples equals the count of $d$-dimensional subspaces of $V_m(q)$.

Minimal base sizes of imprimitive linear groups are explicitly determined.

Certain affine groups satisfy Pyber's conjecture regarding bases.

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.