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Reidemeister Projective Planes

Published online by Cambridge University Press:  20 November 2018

Michael J. Kallaher
Michael J. Kallaher*
Affiliation:
Syracuse University, Syracuse, New York
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By a Reidemeister plane we mean a projective plane having the property that every ternary ring coordinatizing it has associative addition. Finite Reidemeister planes have been investigated by Gleason (2), Liineburg (6), and Kegel and Luneburg (4). In the first paper, Gleason proved that if the order of the plane is a prime power, then it is Desarguesian. Luneburg showed that this is still true if the order is not 60. In the third paper, this last restriction is removed. For infinite planes, the only result is the following theorem due to Pickert (7, p. 301).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1459 - 1464
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

The results in this paper form a part of the author's doctoral dissertation written at Syracuse University under the direction of Professor Erwin Kleinfeld, to whom the author wishes to express his appreciation. This research was supported partially by the U.S. Army Research Office in Durham and partially by a National Aeronautics and Space Administration fellowship at Syracuse University.

References

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2. Gleason, A. M., Finite Fano planes, Amer. J. Math. 78 (1956), 797–807.Google Scholar
3. Hall, Marshall Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
4. Kegel, Otto H. and Lùneburg, Heinz, Ûber die Reidemeisterbedingung. II, Arch. Math. 14 1963), 7–10.Google Scholar
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6. Lüneburg, Heinz, Über die Heine Reidemeisterbedingung, Arch. Math. 12 (1961), 382–384.Google Scholar
7. Pickert, Gunter, Projektive Ebenen (Springer-Verlag, Berlin, 1955).10.1007/978-3-662-00110-3CrossRefGoogle Scholar