On the transitivity of the orthogonal and symplectic groups in projective space

Abstract

1.Introduction. Let Ω be a non-singular quadric in ann-dimensional projective spacePG(n, K) whose coordinate field isK. With respect to Ω the linear subspaces ofPG(n, K) fall into various types: the subspaces of a given type each have the same dimension and the same geometrical kind of quadric section with Ω. Each element of the collineation group Γ preserving Ω takes a subspace into one of the same type, but Γ may divide the subspaces of a given type into several transitivity classes or orbits.