Abstract
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.l.o.g. with translation group K containing G); (3) the dual of N∖P is a translation transversal design (w.l.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 a unified whole.
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References
André, J., ‘Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe’, Math. Z. 60 (1954), 156–186.
Bailey, R. A. and Jungnickel, D., ‘Translation nets and fixed-point-free group automorphisms’ (J. Combin. Theory Ser. A, in press).
Beth, T., Jungnickel, D. and Lenz, H., Design Theory, Cambridge Univ. Press, 1986.
Biliotti, M. and Micelli, G., ‘On translation transversal designs’, Rend. Sem. Mat. Univ. Padova 73 (1985), 217–229.
Bose, R. C. and Bush, K. A., ‘Orthogonal arrays of strength two and three’, Ann. Math. Stat. 23 (1952), 508–524.
Bruck, R. H., ‘Finite nets II. Uniqueness and embedding’, Pacific J. Math. 13 (1963), 421–457.
Bruen, A. A., ‘Unimbeddable nets of small deficiency’, Pacific J. Math. 43 (1972), 51–54.
Dow, S., ‘Transversal-free nets of small deficiency’, Arch. Math. 41 (1983), 472–474.
Evans, A. B., ‘Orthomorphism graphs of groups’, J. Geom. 35 (1989), 66–74.
Hughes, D. R. and Piper, F. C., Projective Planes, Springer, Berlin, Heidelberg, New York, 1973.
Huppert, B., Endliche Gruppen I, Springer, Berlin, Heidelberg, New York, 1967.
Johnson, D., Dulmage, A. M. and Mendelsohn, N. S., ‘Orthomorphisms of groups and orthogonal Latin squares’, Canad. J. Math. 13 (1961), 356–372.
Johnson, N. L., ‘Maximal partial spreads and central groups’ (Preprint).
Jungnickel, D., ‘On difference matrices, resolvable transversal designs, and generalized Hadamard matrices’, Math. Z. 167 (1979), 49–60.
Jungnickel, D., ‘On difference matrices and regular Latin squares’, Abh. Math. Sem. Hamburg 50 (1980), 219–231.
Jungnickel, D., ‘Existence results for translation nets’, in Finite Geometries and Designs. London Math. Soc. Lecture Notes 49 (1981), 172–196, Cambridge Univ. Press.
Jungnickel, D., ‘Maximal partial spreads and translation nets of small deficiency’, J. Algebra 90 (1984), 119–132.
Jungnickel, D., ‘Lateinische Quadrate, ihre Geometrien und ihre Gruppen’, Jahresber. DMV 86 (1984), 69–108.
Jungnickel, D., ‘Existence results for translation nets II’, J. Algebra 122 (1989), 288–298.
Jungnickel, D., ‘Latin squares, their geometries and their groups. A survey’, in D. Ray-Chaudhuri (ed.). Coding Theory and Design Theory Part II. IMA Vol. 21 (1990), Springer, New York, pp. 166–225.
Mann, H. B., ‘The construction of orthogonal Latin squares’, Ann. Math. Stat. 13 (1942), 418–423.
Ostrom, T. G., ‘Nets with critical deficiency’, Pacific J. Math. 14 (1964), 1381–1387.
Ostrom, T. G., ‘Replaceable nets, net collineations, and net extensions’, Canad. J. Math. 18 (1966), 666–672.
Pickert, G., Projektive Ebenen, Springer, Berlin, Heidelberg, New York, 1955.
Pickert, G., Einführung in die endliche Geometrie, Klett, Stuttgart, 1974.
Repphun, K., ‘Geometrische Eigenschaften vollständiger Orthomorphismensysteme von Gruppen’, Math. Z. 89 (1965), 206–212.
Schulz, R.-H., ‘Transversal designs and Hughes-Thompson groups’, Mitt. Math. Sem. Giessen 165 (1984), 185–197.
Schulz, R.-H., ‘Transversal designs and partitions associated with Frobenius groups’, J. Reine Angew. Math. 355 (1985), 153–162.
Schulz, R.-H., ‘On the classification of translation group divisible designs’, Europ. J. Comb. 6 (1985), 369–374.
Sprague, A. P., ‘Translation nets’, Mitt. Math. Sem. Giessen 157 (1982), 46–68.
Vedder, K., ‘Generalised elations’, Bull. London Math. Soc. 18 (1986), 573–579.
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To Helmut Salzmann on the occasion of his 60th birthday
The results of this paper will form part of the first author's doctoral dissertation which is being written under the supervision of the second author.
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Hachenberger, D., Jungnickel, D. Bruck nets with a transitive direction. Geom Dedicata 36, 287–313 (1990). https://doi.org/10.1007/BF00150796
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DOI: https://doi.org/10.1007/BF00150796