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Finite Projective Planes with Affine Subplanes

Published online by Cambridge University Press:  20 November 2018

T. G. Ostrom  and
F. A. Sherk
T. G. Ostrom
Affiliation:
Washington State University and University of Toronto
F. A. Sherk
Affiliation:
Washington State University and University of Toronto
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A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:

If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.

In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 4 , December 1964 , pp. 549 - 559
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Baker, H. F., Principles of Geometry, Vol. V, Cambridge, 1933.Google Scholar
2. Carmichael, R. D., Introduction to the Theory of Groups of Finite Order, Boston, 1937.Google Scholar
3. Coxeter, H. S. M., Introduction to Geometry, New York, 1962.Google Scholar
4. Hall, Marshall Jr., The Theory of Groups, New York, 1959.Google Scholar
5. Neumann, Hanna, On Some Finite Non-Desarguesian Planes, Archiv der Mathematik, VI, 1955, pp. 36–40.Google Scholar