Hostname: page-component-857557d7f7-cmjwd Total loading time: 0 Render date: 2025-11-22T01:40:34.063Z Has data issue: false hasContentIssue false
  • English
  • Français

The existence of Fano subplanes in generalized Hall planes

Published online by Cambridge University Press:  09 April 2009

A. J. Rahilly
A. J. Rahilly
Affiliation:
University of SydneySydney, New South Wales, Australia
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.

One of the best known classes of non-Desarguesian planes is the calss of Hall planes (see Hall [2]). In [6] Hanna Neumann showed that the finite Hall planes of old order possess subplanes of order two (i.e., Fano subplanes)1. Kirkpatrick [5] has considered a type of plane which is generalization of the Hall planes and which he calls generalized Hall planes. In this paper we will give a sufficient condition that a finite generalized Hall plane possesses Fano subplanes. Some examples of odd order planes to which the condition applies shall be exhibited.

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 2 , September 1973 , pp. 234 - 238
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]André, J., ‘Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe’, Math. Z. 60 (1954), 156–186.CrossRefGoogle Scholar
[2]Hall, M. Jr, The Theory of Groups (Macmillan, New York, 1959).Google Scholar
[3]Hughes, D. R., ‘Collineation groups of non-desarguesian planes. I. The Hall Veblen-Wedderburn systems’, Amer. J. Math. 81 (1959), 921–938.CrossRefGoogle Scholar
[4]Johnson, N. L., ‘Translation planes constructed from semifield planes’, Pacific J. Math. (1971), 701–712.CrossRefGoogle Scholar
[5]Kirkpatrick, P. B., ‘Generalization of Hall planes of odd order’, Bull. Austral. Math. Soc. 4 (1971), 205–209.CrossRefGoogle Scholar
[6]Hanna, Neumann, ‘On some finite non-desarguesian planes’, Arch. Math. 6 (1955), 36–40.Google Scholar
[7]Ostrom, T. G., ‘Semi-translation planes’, Trans. Amer. Math. Soc. 111 (1964), 1–18.CrossRefGoogle Scholar