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Completions of Quadrangles in Projective Planes II

Published online by Cambridge University Press:  20 November 2018

Raymond B. Killgrove
Raymond B. Killgrove*
Affiliation:
San Diego State College
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This viewpoint of studying projective planes was given in my previous paper (12). It is discussed in other papers: Hall (4, 6, 7), Maisano (16), Lombardo-Radice (14, 15), Wagner (19). In particular, we consider how to make identifications in the free plane, or how identifications are forced when one begins with a non-degenerate quadrangle and makes free extensions of this quadrangle of a known plane. We shall continue to develop this topic using the notations and definitions of the previous paper (12). We consider the number of subplanes of certain planes, finding exact values in the cases of the known order-nine planes, and deriving a lower bound in a general case. We prove a theorem concerning the structure of all singly generated planes. We give an example to show that this structure is not definitive. Finally, we prove that a specific Walker-Knuth plane is singly generated.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 155 - 165
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Bruck, R. H., Recent advances in the foundations of euclidean plane geometry, Herbert Ellsworth Slaught Memorial Papers, No. 4 (Supplement of the Amer. Math. Monthly, 1955).Google Scholar
2. Carmichael, R. D., Introduction to the theory of groups of finite order (New York, 1937).Google Scholar
3. Gleason, A. M., Finite Fano planes, Amer. J. Math., 78 (1956), 797–807.Google Scholar
4. Hall, M. Jr., Projective planes, Trans. Amer. Math. Soc, 54 (1943), 229–277.Google Scholar
5. Hall, M. Jr., Corrections, Trans. Amer. Math. Soc, 65 (1949), 473–474.Google Scholar
6. Hall, M. Jr., Uniqueness of the projective plane with 57 points, Proc. Amer. Math. Soc, 4 (1953), 912–916.Google Scholar
7. Hall, M. Jr., Correction, Proc Amer. Math. Soc, 5 (1954), 994–997.Google Scholar
8. Hall, M. Jr., Projective planes and related topics (Pasadena, Cal. Tech., April 1954).Google Scholar
9. Hall, M. Jr., theory of groups (New York, 1959).Google Scholar
10. Hall, M. Jr., Swift, J. D., and Killgrove, R., On projective planes of order nine, Math. Tables and Other Aids to Comp., 13 (1959), 233–246.Google Scholar
11. Hughes, D., A class of non-Desarguesian projective planes, Can. J. Math., 9 (1957), 378–388.Google Scholar
12. Killgrove, R. B., Completions of quadrangles in projective planes, Can. J. Math., 16 (1964), 63–76.Google Scholar
13. Knuth, D. E., Finite semifields and projective planes , Thesis, Cal. Tech. (Pasadena, 1963).Google Scholar
14. Lombardo, L.-Radice, Su alcuni carateri dei piani grafici, Rend. Sem. Mat. Padova (1955), 312-345.Google Scholar
15. Lombardo, L.-Radice, Sur la définition de proposition configurationnelle et sur certaines questions algébrogéométriques dans les plans projectifs, Coll. d'Algèbre Supérieure (Bruxelles, 1956), 217-230.Google Scholar
16. Maisano, F., Sulla struttura dei piani liberi di M. Hall, Convegne intern.: Reticoli e géométrie proiettive (Palermo, Messina, 1957), pp. 87–98.Google Scholar
17. Paige, L. J. and Wexler, C., A canonical form for incidence matrices of finite projective planes and their associated Latin squares, Port. Math., 20 (1953), 105–112.Google Scholar
18. Pickert, G., Projektive Ebenen (Berlin, 1955).Google Scholar
19. Wagner, A., On finite non-Desarguesian planes generated by 4 points, Arch. Math., 7 (1956), 23–27.Google Scholar
20. Walker, R. J., Determination of division algebras with 32 elements, Proc. Sym. Exp. Arith., to appear.Google Scholar
21. Zappa, G., Sui gruppi di collineazioni dei piani di Hughes, Boll. Unione Mat. Ital., 12 (1957), 507–516.Google Scholar