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Generalized Hughes Planes

Published online by Cambridge University Press:  20 November 2018

Peter Dembowski
Peter Dembowski*
Affiliation:
Mathematisches Institut Universität Tübingen, West Germany
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The projective planes discovered in 1957 by Hughes [3] were originally described by means of a nearfield F satisfying the following conditions:

(a) F is finite,

(b) the centre and kernel of F coincide,

(c) F is of rank 2 over its kernel.

(The definitions of these terms will be given in § 2; the terminology used throughout the paper is that of [1].)

Rosati [5] showed in 1960 that condition (a) is not necessary, thus constructing the first “infinite Hughes planes”. Condition (b), however, plays an essential part also in Rosati's work.

The aim in this paper is to show that condition (b) is superfluous as well. For the finite case, this has been remarked by Ostrom [4] without proof; here we shall show that a “generalized Hughes plane” can be constructed over any nearfield satisfying condition (c) only.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 481 - 494
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Dembowski, P., Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44 (Springer-Verlag, New York-Berlin-Heidelberg, 1968).Google Scholar
2. Dembowski, P., Gruppenerhaltende quadratische Erweiterungen endlicher desargue s s cher projektiver Ebenen (to appear in Arch. Math.).Google Scholar
3. Hughes, D. R., A class of non-Desarguesian projective planes, Can. J. Math. 9 (1957), 378–388.Google Scholar
4. Ostrom, T. G., Vector spaces and construction of finite projective planes, Arch. Math. 19 (1968), 1–25.Google Scholar
5. Rosati, L. A., Su una generalizzazione dei piani di Hughes, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 29 (1960), 303–308.Google Scholar
6. Rosati, L. A., II gruppo delle collineazioni dei piani di Hughes infiniti, Riv. Mat. Univ. Parma (2) 2 (1961), 19–24.Google Scholar
7. Zassenhaus, H., Über endliche Fastkörper, Abh. Math. Sem. Univ. Hamburg 11 (1935), 187–220.Google Scholar