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{\bf Jing Xu, Michael Giudici, Cai Heng Li and Cheryl E. Praeger}
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{\bf Invariant Relations and Aschbacher Classes of Finite Linear Groups}
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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.
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