Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples (Q267038)

Bases are a fundamental tool in permutation group theory and are extensively used in computational group theory. For a permutation group \(G\) on \(\Omega\), a base is a subset \(B\) of \(\Omega\) with the property that only the identity of \(G\) fixes every point of \(B\). The base size of \(G\) on \(\Omega\), denoted by \(b_\Omega(G)\) or \(b(G)\), is the minimal size of a base for \(G\). In this paper, they study the base sizes of imprimitive linear groups. For a subgroup \(L\) of the symmetric group \(S_{l}\), they determine the minimal base size of \(\mathrm{GL}_d(q)\wr L\) acting on \(V_d(q)^l\) as an imprimitive linear group. This is achieved by computing the number of orbits of \(\mathrm{GL}_d(q)\) on spanning \(m\)-tuples, which turns out to be the number of \(d \)-dimensional subspaces of \(V_m(q)\). They then use these results to prove that for certain families of subgroups \(L\), the affine groups whose stabilisers are large subgroups of \(\mathrm{GL}_d(q)\wr L\) satisfy a conjecture of Pyber concerning bases.