On some projective planes of finite order

Bu çalışmada 2, 3, 4, 5, 7, ve 8 mertebeli projektif düzlemlerin kuruluş metotları incelenmiş, bunların tekliklerine karşılık 9. mertebeden bilinen 4 farklı düzlemin elde edilişi ve bilgisayar araştırmalarına dayalı olarak 10. mertebeden bir projektif düzlemin yokluğu hakkındaki bilgiler derlenerek bir araya getirilmiştir.

Sonlu mertebeli bazı projektif düzlemler üzerine

In this work, construction methods of projective planes of order 2, 3, 4, 5, 7 and 8 are examined. Informations about the obtaining of known four different planes of order 9 and non-existence of a projective plane of order 10 which is obtained according to computer based calculations are collected.

___

  • 1.Tarry, G., Le Probleme des 36 Officiers, Compte Rendu de I'Association Française pour I'Avancement de Science Naturel, 1: 122-123 (1900). 2.Tarry, G., Le Probleme des 36 Officiers, Compte Rendu de I'Association Française pour I'Avancement de Science Naturel, 2: 170-203 (1901). 3.Brack, R.H., Ryser H.J., "The Non-existence of Certain Finite Projective Planes", Canadian Journal of Math., 1: 88-93 (1949). 4.Batten, L.M., Combinatorics of Finite Geometris. Cambridge University Press, 43-44 (1986). 5.Kaya, R., Projektif Geometri.,Anadolu Üniversitesi Yayınları. No:551. Eskişehir, 112-114 (1992). 6.Internet: Projective Plane of order 5. http://www.maths.monash.edu.au/~bpolster/pg5.html. (2004). 7.Beutelspacher, A., 21-6=15: A Connection Between Two Distinguished Geometries, Fachbereich Mathematik der Universitat, Saarstr. 21, D-6500 Mainz, Federal Republic of Germany, 29-40 (1986). 8.internet: Projective Plane of order 4. http://www.win.tue.nl/math/dw/pp/hansc/mathieu/node2.html. (2000). 9.Elkies, N., "Proof the Uniqueness of the Projective Plane of Order 5", elkies@MATH.HARVARD.EDU. (2000). 10.Pierce, W.A., "The Impossibility of Fano's Configuration in a Projective Plane with Eight Points Per Line", Am. Math. Soc. Proc. ,4: 908-912 (1953). 11.Hall/ M. JR., "Uniqueness of the Projective Plane with 57 Points", Am. Math. Soc. Proc. , 4: 912-916 (1953). 12.Hall, M. JR., Correction to Uniqueness of the Projective Plane with 57 Points. Ani. Math. Soc. Proc. , 5: 994-997 (1954). 13.Moufang, R., ,,Zur Straktur der Projektiven Geometrie der Ebene", Math. Ann., 105: 536-601 (1931). 14.Bose, R.C., "On the application of the properties of Galois fields to the construction of hyper-Graeco-Latin squares", Sankhya 3: 323-338 (1938). 15.Stevenson, F. W., Projective Planes. W. H. Freeman and Company, San Francisco, 416s (1972). 16.Laywine, C.F., Mullen, G.L,.,Discrete Mathematics Using Latin Squares, John Wiley&Sons., NewYork. 137-140 (1998). 17.Norton, H.W., "The 7x7 Squares", Ann. Eugenics, 9: 269-307 (1939). 18.Sade, A., "An omission in Norton's list of 7 X7 Squares", Annals Math. Statistics, 22: 306-307 (1951). 19.Hall, M. JR., J.D. Swift and R.J. Walker, Uniqueness of the Projective Plane of Order Eight. Math. Tables Aids. Comput., 10: 186-194 (1956). 20.Hughes, D.R., Piper, F.C., Projective Planes. Springer - Verlag, New York Inc, 196-201 (1973). 21.Room, T.G., Kirkpatnck, V.B.,Miniquaternion Geometry, Cambridge University Press, 176s (1971). 22.Lam, C.W.H., Kolesova, G., Thiel, L. A., "Computer Search for Finite Projective Planes of Order 9", Discrete Mathematics, 92: 187-195 (1991). 23.Lam, C.W.H.,"The Search for a Finite Projective Planes of Order 10.", Computer Science Department, Concordia University, American Math. Monthly, 305-318 (1991).