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Cyclic Affine Planes

Published online by Cambridge University Press:  20 November 2018

A. J. Hoffman
A. J. Hoffman*
Affiliation:
Institute for Advanced Study, Princeton, N.J.
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Let Π be an affine plane which admits a collineation τ such that the cyclic group generated by τ leaves one point (say X) fixed, and is transitive on the set of all other points of Π. Such “cyclic affine planes” have been previously studied, especially in India, and the principal result relevant to the present discussion is the following theorem of Bose [2]: every finite Desarguesian affine plane is cyclic. The converse seems quite likely true, but no proof exists. In what follows, we shall prove several properties of cyclic affine planes which will imply that for an infinite number of values of n there is no such plane with n points on a line.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 295 - 301
Copyright
Copyright © Canadian Mathematical Society 1952

References

1. Baer, R., Projectivities of finite projective planes, Amer. J. Math., vol. 69 (1947), 653–684.Google Scholar
2. Bose, R. C., An affine analogue of Singer's Theorem, J. Indian Math. Soc, vol. 6 (1942), 1–15.Google Scholar
3. Bruck, R. H. and Ryser, H. J., The nonexistence of certain finite projective planes, Can. J. Math., vol. 1 (1949), 88–93.Google Scholar
4. Chowla, S., On difference-sets, J. Indian Math. Soc, vol. 9 (1945), 28–31.Google Scholar
5. Chowla, S. and Ryser, H. J., Combinatorial problems, Can. J. Math., vol. 2 (1950), 93–99.Google Scholar
6. Hall, M., Cyclic projective planes, Duke Math. J., vol. 14 (1947), 1079–1090.Google Scholar
7. Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc, vol. 43 (1938), 377–385.Google Scholar