Invariant Relations and Aschbacher Classes of Finite Linear Groups

Abstract

For a positive integer $k$, a $k$-relation on a set $\Omega$ is a non-empty subset $\Delta$ of the $k$-fold Cartesian product $\Omega^k$; $\Delta$ is called a $k$-relation for a permutation group $H$ on $\Omega$ if $H$ leaves $\Delta$ invariant setwise. The $k$-closure $H^{(k)}$ of $H$, in the sense of Wielandt, is the largest permutation group $K$ on $\Omega$ such that the set of $k$-relations for $K$ is equal to the set of $k$-relations for $H$. We study $k$-relations for finite semi-linear groups $H\leq{\rm\Gamma L}(d,q)$ in their natural action on the set $\Omega$ of non-zero vectors of the underlying vector space. In particular, for each Aschbacher class ${\mathcal C}$ of geometric subgroups of ${\rm\Gamma L}(d,q)$, we define a subset ${\rm Rel}({\mathcal C})$ of $k$-relations (with $k=1$ or $k=2$) and prove (i) that $H$ lies in ${\mathcal C}$ if and only if $H$ leaves invariant at least one relation in ${\rm Rel}({\mathcal C})$, and (ii) that, if $H$ is maximal among subgroups in ${\mathcal C}$, then an element $g\in{\rm\Gamma L}(d,q)$ lies in the $k$-closure of $H$ if and only if $g$ leaves invariant a single $H$-invariant $k$-relation in ${\rm Rel}({\mathcal C})$ (rather than checking that $g$ leaves invariant all $H$-invariant $k$-relations). Consequently both, or neither, of $H$ and $H^{(k)}\cap{\rm\Gamma L}(d,q)$ lie in ${\mathcal C}$. As an application, we improve a 1992 result of Saxl and the fourth author concerning closures of affine primitive permutation groups.