Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-27T09:56:32.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-Desarguesian Projective Plane Geometries Which Satisfy The Harmonic Point Axiom

Published online by Cambridge University Press:  20 November 2018

N. S. Mendelsohn
N. S. Mendelsohn*
Affiliation:
University of Manitoba
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1. Introduction and summary. In her papers (12) and (13) R. Moufang discusses projective plane geometries which satisfy the axiom of the uniqueness of the fourth harmonic point. Her main result is that in such geometries non-homogeneous co-ordinates may be assigned to the points of the plane (except for the “line at infinity”) in such a way that straight lines have equations of the forms aαx + y + β = 0, or x + γ − 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 8 , 1956 , pp. 532 - 562
Copyright
Copyright © Canadian Mathematical Society 1956

References

1. Baer, R., Homogeneity of projective planes. Amer. J. Math. 64 (1942), 137–152.Google Scholar
2. Baker, H. F., Principles of Geometry, vol. 1 (Cambridge, 1929).Google Scholar
3. Bruck et al., R. H., Contributions to geometry, Amer. Math. Monthly, 62 (1955), no. 7, part II.Google Scholar
4. Bruck et al., R. H. and Kleinfeld, E., Structure of alternative division rings, Proc. Amer. Math. Soc. 2 (1951), 878–890.Google Scholar
5. Bruck et al., R. H. and Ryser, H. J., The non-existence of certain finite projective planes, Can. J. Math. 1 (1949), 88–93.Google Scholar
6. Coxeter, H. S. M., The Real Projective Plane (New York, 1949).Google Scholar
7. Hall, M., Uniqueness of the projective plane with 57 points, Proc. Amer. Math. Soc. 4 (1953), 912–916. Correction, 6 (1955).Google Scholar
8. Hall, M., Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229–277.Google Scholar
9. Hua, L. K., Some properties of a S field, Proc. Nat. Aca. Sci. 35 (1949), 533–537.Google Scholar
10. Mendelsohn, N. S., A group theoretic characterization of the general projective collineation group, Trans. Roy. Soc. Can. 40 (1946), Section III, 37–58.Google Scholar
11. Mendelsohn, N. S., Solution to Problem 4062, Amer. Math. Monthly, 51 (1944), 171.Google Scholar
12. Moufang, R., Alternativkörper und der Satz vom vollständigen Vierseit (D, 9), Abh. Math. Sem., Hamburg, 9 (1932), 207–222.Google Scholar
13. Moufang, R., Die Schnittpunktsätze des projektiven speziellen Fönfecksnetzes in ihrer Abhängigkeit voneinander (das A-Netz), Math. Annalen 106 (1932), 755–795.Google Scholar
14. Pickert, G., Projektive Ebenen (Berlin, 1955).Google Scholar
15. Veblen, O. and Weddenburn, J. H. M., Non-Desarguesian and non-Pascalian Geometries, Trans. Amer. Math. Soc. 8 (1907), 379–388.Google Scholar
16. Zassenhaus, H., Ueber endliche Fastkörper, Abh. Math. Sem., Hamburg, 11 (1936), 187–220.Google Scholar