Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-20T08:34:15.908Z Has data issue: false hasContentIssue false

Completions of Quadrangles in Projective Planes

Published online by Cambridge University Press:  20 November 2018

R. B. Killgrove*
Affiliation:
San Diego State College San Diego, California
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.

This paper discusses projective planes from the viewpoint of their classification into singly-generated and non-singly-generated planes. (Singly-generatedness, a property explicit in Hall (6) and Wagner (19), and implicit in Hughes (10), is defined in Section 5.) The elements (points and lines) of a singly-generated plane are expressible in four basic points called a quadrangle. A t'completion procedure" enables us to obtain expressions for the elements of a plane.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Bruck, R. H., Recent advances in the foundations of Euclidean plane geometry, Herbert Ellsworth Slaught Memorial Papers No. 4 (Supplement of Amer. Math. Monthly, 1955).Google Scholar
2. Coxeter, H. S. M., The real projective plane (New York, 1949).Google Scholar
3. Gleason, A. M., Finite Fano planes, Amer. J. Math., 78 (1956), 797807.Google Scholar
4. Hall, Marshall Jr., Projective planes, Trans. Amer. Math. Soc, 54 (1943), 229277.Google Scholar
5. Hall, Marshall Jr., Uniqueness of the projective plane with 57 points, Proc. Amer. Math. Soc, 4 (1953), 912916.Google Scholar
6. Hall, Marshall Jr., Correction to “ Uniqueness of the projective plane with 57 points,” Proc. Amer. Math. Soc, 5 (1954), 994997.Google Scholar
7. Hall, Marshall Jr., The theory of groups (New York, 1959).Google Scholar
8. Hall, M., Swift, J. D., and Killgrove, R. B., On projective planes of order nine, Math. Tables and Other Aids to Comp., 13 (1959), 233246.Google Scholar
9. Hall, M., Swift, J. D., and Walker, R., Uniqueness of the projective plane of order eight, Math. Tables and Other Aids to Comp., 10 (1956), 186194.Google Scholar
10. Hughes, D. R., On homomorphisms of projective planes, Proc Symposium on Applied Math. 10, 4552.Google Scholar
11. Kleinfeld, Erwin, Techniques for enumerating Veblen-Wedderburn systems, J. Assoc, for Comp. Mach., 7 (1960), 330337.Google Scholar
12. Lombardo-Radice, L., Su alcuni caratteri dei piani grafici, Rend. Sem. Univ. Padova, 25 (1955), 312345.Google Scholar
13. Lombardo, L.-Radice, Sur la definition de proposition configurationnelle et sur certaines questions algebrogéomêtriques dans les plans projectifs, Colloque d'Algèbre supérieure (Brussels, 1956).Google Scholar
14. Maisano, F., Sulla struttura dei piani liberi di M. Hall, Convegno internationale, Reticola e géométrie proiettive (Palermo, 1957 Rome, 1958), pp. 87-98.Google Scholar
15. Paige, L. J. and Charles Wexler, A canonical form for incidence matrices of finite projective planes and their associated Latin squares, Portugaliae Math., 12 (1953), 105112.Google Scholar
16. Pickert, Guenter, Projektive Ebenen (Berlin, 1955).Google Scholar
17. Segre, B., Lezioni di geometria moderna, vol. I (Bologna, 1948).Google Scholar
18. Tarry, G., Le problème des 36 officiers, C.R. Assoc. Française pour l'Avancement de la Science Naturelle, I (1900), 122-123; II, 170-203.Google Scholar
19. Wagner, A., On finite non-desarguesian planes generated by 4 points, Arch. d. Math. 7 (1956), 2327.Google Scholar
20. Wagner, A., On projective planes transitive on quadrangles, J. London Math. Soc, 33 (1958), 2533.Google Scholar
21. Zappa, Guido, Piani grafici a caratteristica 3, Ann. Mat. Pura Appl., 49 (1960), 157166.Google Scholar