Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T07:55:33.705Z Has data issue: false hasContentIssue false
  • English
  • Français

A Double-Infinity Configuration

Published online by Cambridge University Press:  09 April 2009

D. W. Barnes
D. W. Barnes
Affiliation:
Department of Pure Mathematics University of Sydney
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The double-six configuration in classical 3-dimensional projective geometry has been discussed by a number of authors. It consists of two sets a1,…, a6 and b1, …, b6 of six lines such that no two lines of the same set intersect, and ai meets bi if and only if i †j. The existence of a doublesix in the 3-dimensional projective geometry over a field F has been proved by Hirschfeld in [2] for all fields F except those of 2, 3 and 5 elements. For an arbitrary 3-dimensional projective geometry in which the number of points on a line is at least 5 but is not 6, the existence of a double-six follows from the fact that the geometry is a geometry over a division ring D with a subfield F satisfying the conditions of Hirschfeld's theorem.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 1-2 , February 1969 , pp. 209 - 210
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Baker, H. F., Principles of Geometry, Vol. III (C.U.P., 1923).Google Scholar
[2]Hirschfeld, J. W. P., Ph. D. Thesis, Edinburgh (1965).Google Scholar
[3]Neumann, B. H., ‘On ordered division rings’, Trans. Amer. Math. Soc. 66 (1949) 202–252.CrossRefGoogle Scholar
[4]Neumann, B. H., ‘On ordered groups’ Amer. J. Math. 71 (1949) 1–18.CrossRefGoogle Scholar