Invariant relations and Aschbacher classes of finite linear groups. (Q665743)

Summary: 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 \(H\leq\Gamma\text{L}(d,q)\) 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 \(\Gamma\text{L}(d,q)\), we define a subset \(\text{Rel}(\mathcal C)\) 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 \(\text{Rel}(\mathcal C)\), and (ii) that, if \(H\) is maximal among subgroups in \(\mathcal C\), then an element \(g\in\Gamma\text{L}(d,q)\) lies in the \(k\)-closure of \(H\) if and only if \(g\) leaves invariant a single \(H\)-invariant \(k\)-relation in \(\text{Rel}(\mathcal C)\) (rather than checking that \(g\) leaves invariant all \(H\)-invariant \(k\)-relations). Consequently both, or neither, of \(H\) and \(H^{(k)}\cap\Gamma\text{L}(d,q)\) 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.