Subgroups of Classical Groups that are Transitive on Subspaces

Abstract

For each finite classical group G, we classify the subgroups of G which act transitively on a G-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the classification of the maximal factorisations of almost simple groups. As a first application of these results we classify all point-transitive subgroups of automorphisms of finite thick generalised quadrangles. * ε ′ ∈ {+, −} and the two classes N ε ′ k can be distinguished by the discriminant of the restriction of the quadratic form. * * These subspaces correspond to quadratic forms on the symplectic space V /V ⊥ of type ε ′ ∈ {+, −}.