On some incidence structures

Abstract

In this paper,suggested by André's papers ([2), [3]), we construct geometrical structures (X,ℒ,//}) where X is a finite set of points, ℒ is a set of lines, and // is an equivalence relation on ℒ. These constructions are made starting with a finite and not empty set X and a permutation group G which is 2-transitive on X and such that the stabilizer of two distinct points of X is different from the identical subgroup. We look for conditions such that the structure (X, ℒ) is a (3,q)-Steiner system. We remember that a (3,q)-Steiner system is a pair (X,B), where X is a set of elements (called points), B is a system of subsets of X (called blocks), such that:

• i) every block contains q points exactly;

• ii) given three distinct points x,y,z of X, there is exactly one subset of X belonging to B and containing x,y,z.

At the end we construct such a system with the help of a nearskewfield (according to Zassenhaus [7], [8]).

Zusammenfassung

Sei G eine auf X zweifach transitive Permutationsgruppe und sei der Stabilisator G_xy≠{1} ($-$x,y ∈ X,x≠y). Jeder solchen Permutationsgruppe G auf X, ordnen wir eine Inzidenzstruktur zu,deren Punkte die Elemente aus X und deren Blöcke,hier Linien genannt, die Punktmengen der Form f(x,y,z)={x,y}∪G_xy(z) ($-$x,y,z ∈ X, x≠y,z; z≠y) sind; die Inzidenz ist als mengentheoretische Inklusion “∈” gegeben.

Wir konstruieren ein (3,q)-Steiner System, mit Hilfe eines Fastkörpers (im Sinne von Zassenhaus [7], [8]).