Bruck nets with a transitive direction: to Helmut Salzmann on the occasion of his 60th birthday

  • 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.show moreshow less

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Metadaten
Author:Dirk HachenbergerORCiDGND, Dieter JungnickelORCiDGND
URN:urn:nbn:de:bvb:384-opus4-8216
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/965
ISSN:1572-9168OPAC
Parent Title (English):Geometriae Dedicata
Publisher:Springer
Place of publication:Dordrecht [u.a.]
Type:Article
Language:English
Year of first Publication:1990
Publishing Institution:Universität Augsburg
Release Date:2008/06/18
Tag:collineation groups; Bruck nets; transitive direction; translation net; group theory; projective geometry
GND-Keyword:Translation <Mathematik>; Gruppentheorie; Projektive Geometrie; Kollineation
Volume:36
Issue:2/3
First Page:287
Last Page:313
DOI:https://doi.org/10.1007/BF00150796
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Diskrete Mathematik, Optimierung und Operations Research
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik