Transitivity in finite general linear groups

Abstract 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 {\, GL\, } (n, q) GL (n, q) acting transitively on flag-like structures, which are common generalisations of t -dimensional subspaces of Fqⁿ F q n and bases of t -dimensional subspaces of Fqⁿ F q n. We give structural characterisations of transitive subsets of {\, GL\, } (n, q) GL (n, q) using the character theory of {\, GL\, } (n, q) GL (n, q) and interpret such subsets as designs in the conjugacy class association scheme of {\, GL\, } (n, q) GL (n, q). In particular we generalise a theorem of Perin on subgroups of {\, GL\, } (n, q) GL (n, q) acting transitively on t -dimensional subspaces. We survey transitive subgroups of {\, GL\, } (n, q) GL (n, q), showing that there is no subgroup of {\, GL\, } (n, q) GL (n, q) with 1 1 t n acting transitively on t -dimensional subspaces unless it contains {\, SL\, } (n, q) 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 {\, GL\, } (n, q) GL (n, q) that are transitive on linearly independent t -tuples of Fqⁿ F q n, which also shows the existence of nontrivial subsets of {\, GL\, } (n, q) 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 {\, GL\, } (n, q) GL (n, q). Many of our results can be interpreted as q -analogs of corresponding results for the symmetric group.