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BRUCK NETS AND PARTIAL SHERK PLANES

Published online by Cambridge University Press:  19 June 2017

JOHN BAMBERG
Affiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, W.A. 6009, Australia email John.Bamberg@uwa.edu.au
JOANNA B. FAWCETT
Affiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, W.A. 6009, Australia email j.b.fawcett@dpmms.cam.ac.uk
JESSE LANSDOWN*
Affiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, W.A. 6009, Australia email Jesse.Lansdown@research.uwa.edu.au

Abstract

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

Type
Research Article
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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Footnotes

The first author acknowledges the support of the Australian Research Council (ARC) Future Fellowship FT120100036. The second author acknowledges the support of the ARC Discovery Grant DP130100106. The third author acknowledges the support of the ARC Discovery Grant DP0984540.

Current address: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

References

Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959).CrossRefGoogle Scholar
Baer, R., ‘Polarities in finite projective planes’, Bull. Amer. Math. Soc. (N.S.) 52 (1946), 7793.CrossRefGoogle Scholar
Bruck, R. H., ‘Finite nets. II. Uniqueness and imbedding’, Pacific J. Math. 13 (1963), 421457.CrossRefGoogle Scholar
Bruen, A., ‘Unimbeddable nets of small deficiency’, Pacific J. Math. 43 (1972), 5154.CrossRefGoogle Scholar
Dembowski, P., ‘Finite geometries’, in: Classics in Mathematics (Springer, Berlin, 1997), reprint of the 1968 original.Google Scholar
Sherk, F. A., ‘Finite incidence structures with orthogonality’, Canad. J. Math. 19 (1967), 10781083.CrossRefGoogle Scholar

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