@article{53534,
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 $\operatorname{GL}(n,q)$ 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
$\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$
and interpret such subsets as designs in the conjugacy class association
scheme of $\operatorname{GL}(n,q)$. In particular we generalise a theorem of
Perin on subgroups of $\operatorname{GL}(n,q)$ acting transitively on
$t$-dimensional subspaces. We survey transitive subgroups of
$\operatorname{GL}(n,q)$, showing that there is no subgroup of
$\operatorname{GL}(n,q)$ with $1