Transitivity in finite general linear groups

It is known that the notion of a transitive subgroup of a permutation group G extends naturally to subsets of G. We consider subsets of the general linear group \({{\,\textrm{GL}\,}}(n,q)\) acting transitively on flag-like structures, which are common generalisations of t-dimensional subspaces of \(\mathbb {F}_q^n\) and bases of t-dimensional subspaces of \(\mathbb {F}_q^n\). We give structural characterisations of transitive subsets of \({{\,\textrm{GL}\,}}(n,q)\) using the character theory of \({{\,\textrm{GL}\,}}(n,q)\) and interpret such subsets as designs in the conjugacy class association scheme of \({{\,\textrm{GL}\,}}(n,q)\). In particular we generalise a theorem of Perin on subgroups of \({{\,\textrm{GL}\,}}(n,q)\) acting transitively on t-dimensional subspaces. We survey transitive subgroups of \({{\,\textrm{GL}\,}}(n,q)\), showing that there is no subgroup of \({{\,\textrm{GL}\,}}(n,q)\) with \(1<t<n\) acting transitively on t-dimensional subspaces unless it contains \({{\,\textrm{SL}\,}}(n,q)\) or is one of two exceptional groups. On the other hand, for all fixed t, we show that there exist nontrivial subsets of \({{\,\textrm{GL}\,}}(n,q)\) that are transitive on linearly independent t-tuples of \(\mathbb {F}_q^n\), which also shows the existence of nontrivial subsets of \({{\,\textrm{GL}\,}}(n,q)\) that are transitive on more general flag-like structures. We establish connections with orthogonal polynomials, namely the Al-Salam–Carlitz polynomials, and generalise a result by Rudvalis and Shinoda on the distribution of the number of fixed points of the elements in \({{\,\textrm{GL}\,}}(n,q)\). Many of our results can be interpreted as q-analogs of corresponding results for the symmetric group.