@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 $1<t<n$ acting transitively on $t$-dimensional
subspaces unless it contains $\operatorname{SL}(n,q)$ or is one of two
exceptional groups. On the other hand, for all fixed $t$, we show that there
exist nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on
linearly independent $t$-tuples of $\mathbb{F}_q^n$, which also shows the
existence of nontrivial subsets of $\operatorname{GL}(n,q)$ 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 $\operatorname{GL}(n,q)$. Many of our results can be
interpreted as $q$-analogs of corresponding results for the symmetric group.}},
  author       = {{Ernst, Alena and Schmidt, Kai-Uwe}},
  journal      = {{Mathematische Zeitschrift}},
  number       = {{45}},
  title        = {{{Transitivity in finite general linear groups}}},
  doi          = {{10.1007/s00209-024-03511-x}},
  volume       = {{307}},
  year         = {{2024}},
}

@article{53533,
  author       = {{Ernst, Alena and Schmidt, Kai-Uwe}},
  issn         = {{0305-0041}},
  journal      = {{Mathematical Proceedings of the Cambridge Philosophical Society}},
  keywords     = {{General Mathematics}},
  number       = {{1}},
  pages        = {{129--160}},
  publisher    = {{Cambridge University Press (CUP)}},
  title        = {{{Intersection theorems for finite general linear groups}}},
  doi          = {{10.1017/s0305004123000075}},
  volume       = {{175}},
  year         = {{2023}},
}