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 $\operatorname{GL}(n,q)$ 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.
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