On the orbits of collineation groups

Abstract

In this paper we consider some results on the orbits of groups of collineations, or, more generally, on the point and block classes of tactical decompositions, on symmetric balanced incomplete block designs (symmetric BIBD = (v, k, 2)system=finite 2-plane), and we consider generalizations to (not necessarily symmetric) BIBD and other combinatorial designs. The results are about the number of point and block classes (or orbits, i.e. sets of transitivity) and the numbers of elements in these classes. In Sections 2, 3 and 4 below we exhibit the key role of the rank of the incidence matrix of a design, while the remainder of the paper uses more specific properties of the incidence relations. Included in Section 2 is a simple new proof of the theorem of DEMBOWSKI [7] on the equality of the numbers of