Open Access

Dirk Hachenberger, Dieter Jungnickel

• We start the systematic investigation of the geometric properties and the collineation groups of Bruck nets N with a transitive direction (i.e. with a group G of central translations acting transitively on each line of a given parallel class P). After reviewing some basic properties of such nets (in particular, their connection to difference matrices), we shall consider the problem of what can be said if either N or G admits an interesting extension. Specifically, we shall handle the following four situations: (1) there is a second transitive direction; (2) N is a translation net (w.1.o.g. with translation group K containing G); (3) the dual of N \ P is a translation transversal design (w.1.o.g. with translation group K containing G); (4) N admits a transversal (and can then in fact be extended by adding a further parallel class). Our study of these problems will yield interesting generalizations of known concepts (e.g. that of a fixed-point-free group automorphism) and results (forWe start the systematic investigation of the geometric properties and the collineation groups of Bruck nets N with a transitive direction (i.e. with a group G of central translations acting transitively on each line of a given parallel class P). After reviewing some basic properties of such nets (in particular, their connection to difference matrices), we shall consider the problem of what can be said if either N or G admits an interesting extension. Specifically, we shall handle the following four situations: (1) there is a second transitive direction; (2) N is a translation net (w.1.o.g. with translation group K containing G); (3) the dual of N \ P is a translation transversal design (w.1.o.g. with translation group K containing G); (4) N admits a transversal (and can then in fact be extended by adding a further parallel class). Our study of these problems will yield interesting generalizations of known concepts (e.g. that of a fixed-point-free group automorphism) and results (for affine and projective planes). We shall also see that a wide variety of seemingly unrelated results and constructions scattered in the literature are in fact closely related and should be viewed as part of unified whole.…