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On Projective Planes by CAFER C¸ALIS¸KAN A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, Florida May 2010 Copyright by CAFER C¸ALIS¸KAN 2010 ii Acknowledgements First of all, I would like to express my gratitude to my advisor Spyros S. Magliveras. Without his expertise, never-ending support and guidance I could not have written this thesis. I also thank the members of my committee – Professors Ron C. Mullin, Lee Klingler and Rainer Steinwandt – who were more than generous with their expertise and precious time. A very special thanks goes to Heinrich Niederhausen, who has supported me in many ways. I would also like to use this opportunity to thank the faculty members and my colleagues at the Department of Mathematical Sciences at Florida Atlantic University. I would like to thank my family for all the support they provided and in particular, I must acknowledge my dearest, Burcu. Without her love and understanding, I would not have finished this manuscript. Boca Raton, Florida Cafer C¸alıs¸kan March 30th; 2010 iv Abstract Author: CAFER C¸ALIS¸KAN Title: On Projective Planes Institution: Florida Atlantic University Dissertation Advisor: Dr. Spyros S. Magliveras Degree: Doctor of Philosophy Year: 2010 This work was motivated by the well-known question: “Does there exist a non- desarguesian projective plane of prime order?” For a prime p < 11, there is only the pap- pian plane of order p. Hence, such planes are indeed desarguesian. Thus, it is of interest to examine whether there are non-desarguesian planes of order 11. A suggestion by As- cher Wagner in 1985 was made to Spyros S. Magliveras: “Begin with a non-desarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations, and check whether one of these is non-desarguesian.” In this manuscript we use a group-theoretic methodology to determine the subplane structures of some non-desarguesian planes. In particular, we determine orbit representatives of all proper Q-subplanes both of a Veblen-Wedderburn (VW) plane Π of order 121 and of the Hughes plane Σ of order 121, under their full collineation groups. In Π, there are 13 orbits of Baer subplanes, all of which are desarguesian, and approximately 3000 orbits of Fano subplanes. In Σ, there are 8 orbits of Baer subplanes, all of which are desarguesian, 2 orbits of subplanes of order 3, and at most 408; 075 distinct Fano subplanes. In addition to the above results, we also study the subplane structures of some non-desarguesian planes, such as the Hall plane of v order 25, the Hughes planes of order 25 and 49, and the Figueora planes of order 27 and 125. A surprising discovery by L. Puccio and M. J. de Resmini was the existence of a plane of order 3 in the Hughes plane of order 25. We generalize this result, showing that there are subplanes of order 3 in the Hughes planes of order q2, where q is a prime power and q ≡ 5 (mod 6). Furthermore, we analyze the structure of the full collineation groups of certain Veblen- Wedderburn (VW) planes of orders 25, 49 and 121, and discuss how to recover the planes from their collineation groups. vi Dedication To my love Burcu Sayım C¸alıs¸kan. “Science is the most genuine guide in life.” – Mustafa Kemal (1881 – 1931) On Projective Planes List of Tables............................................................................ ix List of Figures ...........................................................................x 1 Introduction .............................................................................1 2 Preliminaries ............................................................................4 3 Some non-desarguesian planes ...................................................... 21 3.1 A VW plane π .................................................................... 21 3.2 The Hall plane & .................................................................. 22 3.3 The Hughes and Figueroa planes ................................................ 23 3.3.1 The Hughes plane σ ..................................................... 23 3.3.2 The Figueroa plane ξ .................................................... 24 4 Subplane structures of planes ....................................................... 26 4.1 Closures of quadrangles and their orbits ........................................ 26 4.2 Subplanes of some non-desarguesian planes of small order ................... 29 4.3 Some non-desarguesian planes of order 121 .................................... 31 4.3.1 π of order 121 (Π)....................................................... 32 4.3.2 σ of order 121 (Σ) ....................................................... 35 4.4 Subplanes of order 3 in the Hughes Planes ..................................... 37 vii 4.4.1 Case: q ≡ 5 (mod 12) ................................................... 39 4.4.2 Case: q ≡ 11 (mod 12).................................................. 40 4.5 Further substructures of the Hughes Planes..................................... 42 5 Reconstructing planes from their groups........................................... 45 5.1 The structure of Gπ .............................................................. 45 5.2 Reconstruction from Gπ ......................................................... 47 5.2.1 Counting Principle....................................................... 47 5.2.2 Reconstruction ........................................................... 48 List of Notations ....................................................................... 51 Appendices ............................................................................. 52 A VW plane of order 25 (α)............................................................. 52 B VW plane of order 49 (β)............................................................. 53 C VW plane of order 121 (Π)........................................................... 55 Bibliography ........................................................................... 57 viii List of Tables 2.1 Projective planes of small order ..................................................... 16 ix List of Figures 2.1 Desargues’ configuration ..............................................................8 2.2 Pappus’ configuration ..................................................................9 2.3 Coordinatization of a projective plane .............................................. 16 4.1 Finding the orbit representatives of the point-sets of planes in Dt. ................ 28 4.2 The full collineation group GΠ ...................................................... 32 4.3 The full collineation group GΣ ...................................................... 36 5.1 The full collineation group Gπ ...................................................... 46 5.2 Counting principle ................................................................... 47 5.3 Representing certain points and lines by involutions ............................... 48 5.4 Determining lines of Type I by certain group elements of order p ................. 49 5.5 Determining lines of Type I by certain group elements of order p ................. 50 x Chapter 1 Introduction A question that is still outstanding in finite geometry is whether all finite projective planes of prime order are desarguesian. For a prime p < 11, there is only the pappian plane of order p [3]. Hence, such planes are indeed desarguesian. Thus, it is still of interest to examine whether there are non-desarguesian planes of order 11. A suggestion by Ascher Wagner in 1985 was made to Spyros S. Magliveras. It is to begin with a non-desarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations and check whether one of these is non-desarguesian. In 1985, the computational complexity of the problem was certainly high enough to make the problem rather intractable, even for k = 2. Since there are planes which are not generated by quadrangles, we reserve the term Q-plane (Q-subplane) for a plane (sub- plane) generated by a quadrangle. In this manuscript, we determine all proper subplanes of some well-known non-desarguesian planes of small order and all proper Q-subplanes both of a Veblen-Wedderburn (VW) plane Π of order 121 and the Hughes plane Σ of order 121 up to collineations, paying special attention to the Baer subplanes of Π and Σ. In particular, there are 19 orbits of Fano subplanes and 6 orbits of subplanes of order 5 in the Hall plane of order 25. The Hughes plane of order 25 has 56 orbits of Fano subplanes, only 1 orbit of subplanes of order 3 and 5 orbits of subplanes of order 5. There are 3524 orbits of Fano subplanes and 59 orbits of subplanes of order 3 in the Figueroa plane of 1 order 27. None of these planes, namely the Hall plane of order 25, the Hughes plane of order 25 or the Figueroa plane of order 27, have a subplane of order 4. We also show that the Hughes plane of order 49 does not have subplanes of order 3. Moreover, the VW plane Π of order 121 contains only Fano subplanes and desargue- sian Baer subplanes as proper Q-subplanes. In Π, there are 13 orbits of subplanes of order 11 all of which are desarguesian and approximately 3000 orbits of Fano subplanes. Subplanes of order 4 or 8 may still exist, since these projective planes are not Q-planes. In a desarguesian plane of order 9, every quadrangle generates a subplane of order 3. However, Π does not have a subplane of order 3. This implies that
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