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BRUCK NETS AND PARTIAL SHERK PLANES

  • JOHN BAMBERG (a1), JOANNA B. FAWCETT (a2) and JESSE LANSDOWN (a3)
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.

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Corresponding author
Jesse.Lansdown@research.uwa.edu.au
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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

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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