Invariant Relations and Aschbacher Classes of Finite Linear Groups

For a positive integer k, a k-relation on a set Ω is a non-empty subset ∆ of the k-fold Cartesian product Ω; ∆ is called a k-relation for a permutation group H on Ω if H leaves ∆ invariant setwise. The k-closure H of H, in the sense of Wielandt, is the largest permutation group K on Ω 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 ≤ ΓL(d, q) in their natural action on the set Ω of non-zero vectors of the underlying vector space. In particular, for each Aschbacher class C of geometric subgroups of ΓL(d, q), we define a subset Rel(C) of k-relations (with k = 1 or k = 2) and prove (i) that H lies in C if and only if H leaves invariant at least one relation in Rel(C), and (ii) that, if H is maximal among subgroups in C, then an element g ∈ ΓL(d, q) lies in the k-closure of H if and only if g leaves invariant a single H-invariant k-relation in Rel(C) (rather than checking that g leaves invariant all H-invariant k-relations). Consequently both, or neither, of H and H ∩ ΓL(d, q) lie in C. As an application, we improve a 1992 result of Saxl and the fourth author concerning closures of affine primitive permutation groups.