Linear frames as orbits of projective frames

A multidimensional projective space with a marked point (center) is considered. On the manifold of projective frames of the given space that are adapted to the center, the action of the stabilizer of the center of the group of projective transformations is defined. We prove that linear frames, i.e., bases of the tangent vector space of the projective space at its center, can be identified with orbits of the adapted projective frames with respect to the action of the kernel of the epimorphism of Lie groups, which assigns to each transformation from the stabilizer its differential at the center. Using a multidimensional generalization of the Desargues theorem, we obtain a criterion for two adapted projective frames to belong to the same orbit.