---
res:
bibo_abstract:
- "It is known that the notion of a transitive subgroup of a permutation group\r\n$G$
extends naturally to subsets of $G$. We consider subsets of the general\r\nlinear
group $\\operatorname{GL}(n,q)$ acting transitively on flag-like\r\nstructures,
which are common generalisations of $t$-dimensional subspaces of\r\n$\\mathbb{F}_q^n$
and bases of $t$-dimensional subspaces of $\\mathbb{F}_q^n$. We\r\ngive structural
characterisations of transitive subsets of\r\n$\\operatorname{GL}(n,q)$ using
the character theory of $\\operatorname{GL}(n,q)$\r\nand interpret such subsets
as designs in the conjugacy class association\r\nscheme of $\\operatorname{GL}(n,q)$.
In particular we generalise a theorem of\r\nPerin on subgroups of $\\operatorname{GL}(n,q)$
acting transitively on\r\n$t$-dimensional subspaces. We survey transitive subgroups
of\r\n$\\operatorname{GL}(n,q)$, showing that there is no subgroup of\r\n$\\operatorname{GL}(n,q)$
with $1