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-Homogeneity of Projective Planes and Polarities

Published online by Cambridge University Press:  20 November 2018

W. Jónsson*
Affiliation:
University of Manitoba
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In (1) Baer introduced the concept of (C, γ)-transitivity and (C, γ)- homogeneity. A projective plane (see (5) for the requisite definitions and axioms) is (C, γ)-transitive if, given an ordered pair (P1, P2) of points collinear with C but distinct from C and not on γ, there is a collineation which maps P1 into P2 and leaves fixed every point on γ as well as every line through C. A projective plane is (C, γ)-homogeneous for a non-incident point-line-pair if it is (C, γ) transitive and there is a correlation which maps every line through C into its intersection with γ and every point on γ into its join with C.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Baer, R., Homogeneity of projective planes, Amer. J. Math., 64 (1942), 137–52.Google Scholar
2. Barlotti, A., Le possibili configurazioni del sistema delle coppie punto-retta (A, a) per oui un piano grafico risulta (A, a) transitivo, Boll. U.M.I., 12 (1957), 212–26.Google Scholar
3. Lenz, H., Kleiner Desargues s cher Satz und Dualität in projektiven Ebenen, Jber. Dtsch. Math. Ver., 57 (1954), 2031.Google Scholar
4. Jönsson, W. J., Transitivitdt und Homogenität projektiver Ebenen, Math. Z.7 80 (1963), 269–92.Google Scholar
5. Pickert, G., Projektive Ebene (Berlin, 1955).Google Scholar