Transitivity in finite general linear groups

Abstract

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