Skip to main content Accessibility help
×
Home

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.

Copyright

Corresponding author

Footnotes

Hide All

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

Footnotes

References

Hide All
[1] Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959).
[2] Baer, R., ‘Polarities in finite projective planes’, Bull. Amer. Math. Soc. (N.S.) 52 (1946), 7793.
[3] Bruck, R. H., ‘Finite nets. II. Uniqueness and imbedding’, Pacific J. Math. 13 (1963), 421457.
[4] Bruen, A., ‘Unimbeddable nets of small deficiency’, Pacific J. Math. 43 (1972), 5154.
[5] Dembowski, P., ‘Finite geometries’, in: Classics in Mathematics (Springer, Berlin, 1997), reprint of the 1968 original.
[6] Sherk, F. A., ‘Finite incidence structures with orthogonality’, Canad. J. Math. 19 (1967), 10781083.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Related content

Powered by UNSILO

BRUCK NETS AND PARTIAL SHERK PLANES

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

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.