PERSPECTIVITIES AND THE LITTLE PROJECTIVE GROUP

Abstract

This chapter discusses projective spaces and its perceptivities. Perceptivities, and especially elations, assume a central role in investigation, and it turns out that very often the group generated by all elations—the little projective group—is already generated by quite a small number of elations. This may probably be ascribed to the fact that the little projective group in these cases is simple. In the case of projective planes, the existence of this group then enables to deduce that the planes are alternative. By a projective space will be understood a system of points and lines connected by axioms of incidence in the usual manner. Points are subspaces of dimension zero, lines are subspaces of dimension one, and if the space has dimension k, the subspaces of dimension k — 1 will be called hyperplanes. A projective space containing only a finite number of points will be called finite. If a finite projective space has n + 1 points on a line, and hence every fine, the space will be said to have order n.