BRUCK NETS AND PARTIAL SHERK PLANES

John Bamberg, Joanna B. Fawcett, Jesse Lansdown

    Research output: Contribution to journalArticle

    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.

    Original languageEnglish
    Pages (from-to)1-12
    Number of pages12
    JournalJournal of the Australian Mathematical Society
    DOIs
    Publication statusPublished - 1 Feb 2018

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