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