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\,(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 $$1<t<n$$ 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.