On the Orbits of Collineation Groups

1. Introduction 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 point and block classes for a tactical decomposition of a symmetric BIBD (for the orbits of a group of collineations the equality is a consequence of a result of BRAUER [4, p. 934], and was proved again by PARKER [12] and HUGHES [10]). Our proof generalizes the equality to a pair of inequalities for non-symmetric designs. In Section 3 we consider transitive groups of collineations, and in Section 4, cyclic groups. We use an integral matrix congruence in Section 5 to prove a type of symmetry for tactical decompositions on symmetric designs. In particular for primes not dividing n =k-2 we prove that such a decomposition is p-symmetric, i.e. the point and block classes can be paired so that paired classes have numbers of elements divisible by the same powers of p; this generalizes other results of DEMBOWSKI [7]. In Section 6 these results are used in conjunction with the theory of rational congruence of quadratic forms to obtain number-theoretic conditions on the numbers of elements in the point and block classes, generalizing the result of LENZ [11]. Finally, in Section 7 we generalize the result of Section 5 on p-symmetry to some inequalities for n0n-symmetric designs. 2. One-Sided Tactical Decompositions For any (generalized) incidence structure, i.e. set of points and blocks with an incidence relation between points and blocks, a tactical decomposition is a partition of the points into point classes and of the blocks into block classes