Bruck nets with a transitive direction
This paper gives a systematic investigation of the geometric properties and the collineation groups of finite Bruck nets \(N=(V,L,\|)\) with a transitive direction, i.e. with a group G of central translations acting transitively on each line of a given parallel class P. Basic properties of such nets are their connection to difference matrices [see the second author, Jahresber. Dtsch. Math.-Ver. 86, No.3, 69-108 (1984; Zbl 0581.05014)] and to sets of pairwise orthogonal orthomorphisms. Among other things it is a remarkable geometrical result that central translation may be considered to be the natural generalization of the concept of translations from affine plane to Bruck nets. After some basic properties the authors consider the problem of what can be said if either N or G admits an interesting extension. They study the following four situations: (1) There is a second transitivity direction; (2) N is a translation net; (3) The dual of \(N\setminus P\) is a translation transversal design; (4) N admits a transversal (and can then in fact be extended by adding a further parallel class). The investigations of these problems result in some essentially equivalent concepts and conditions. They yield interesting generalizations of known concepts (e.g. that of a fixed-point-free group automorphism) and results for affine and projective planes. New results of this paper contain parts of the first author's doctoral dissertation (U. Gießen). For details see also there.
scientific article; zbMATH DE number 4192525
scientific article