Translation Planes

ISBN-13: 9783642674143 / Angielski / Miękka / 2011 / 278 str.

H. L. Neburg

Opis

Wir unterhielten uns einmal dariiber, daB man sich in einer fremden Sprache nur unfrei ausdriicken kann und im Zweifelsfall lieber das sagt, was man richtig und einwandfrei zu sagen hofft, als das, was man eigentlich sagen will. Molnar nickte bestatigend: "Es ist sehr traurig", resiimierte er. "Ich habe oft mitten im Satz meine Weltanschauung andem miissen ..." Friedrich Torberg, Die Tante Jolesch The last two decades have witnessed great progress in the theory of translation planes. Being interested in, and having worked a little on this subject, I felt the need to clarify for myself what had been happening in this area of mathematics. Thus I lectured about it for several semesters and, at the same time, I wrote what is now this book. It is my very personal view of the story, which means that I selected mainly those topics I had touched upon in my own investigations. Thus finite translation planes are the main the~ of the book. Infinite translation planes, however, are not completely disregarded. As all theory aims at the mastering of the examples, these play a central role in this book. I believe that this fact will be welcomed by many people. However, it is not a beginner's book of geometry. It presupposes consider- able knowledge of projective planes and algebra, especially group theory. The books by Gorenstein, Hughes and Piper, Huppert, Passman, and Pickert mentioned in the bibliography will help to fill any gaps the reader may have.

Spis treści

I Introduction.- 1. André’s Description of Translation Planes.- 2. An Alternative Description of Translation Planes.- 3. Homologies and Shears of Translation Planes.- 4. A Characterization of Pappian Planes.- 5. Quasifields.- II Generalized André Planes.- 6. Some Number Theoretic Tools.- 7. Finite Nearfield Planes.- 8. The Nearfield Plane of Order 9.- 9. Generalized André Planes.- 10. Finite Generalized André Planes.- 11. Homologies of Finite Generalized André Planes.- 12. The André Planes.- 13. The Hall Planes.- 14. The Collineation Group of a Generalized André Plane.- III Rank-3-Planes.- 15. Line Transitive Affine Planes.- 16. Affine Planes of Rank 3.- 17. Rank-3-Planes with an Orbit of Length 2 on the Line at Infinity.- 18. The Planes of Type R*p.- 19. The Planes of Type F*p.- 20. Exceptional Rank-3-Planes.- IV The Suzuki Groups and Their Geometries.- 21. The Suzuki Groups S(K,?).- 22. The Simplicity of the Suzuki Groups.- 23. The Lüneburg Planes.- 24. The Subgroups of the Suzuki Groups.- 25. Möbius Planes.- 26. The Möbius Planes Belonging to the Suzuki Groups.- 27. S(q) as a Collineation Group of PG(3, q).- 28. S(q) as a Collineation Group of a Plane of Order q2.- 29. Geometric Partitions.- 30. Rank-3-Groups.- 31. A Characterization of the Lüneburg Planes.- V Planes Admitting Many Shears.- 32. Unitary Polarities of Finite Desarguesian Projective Planes and Their Centralizers.- 33. A Characterization of A5.- 34. A Characterization of Galois Fields of Odd Characteristic.- 35. Groups Generated by Shears.- VI Flag Transitive Planes.- 36. The Uniqueness of the Desarguesian Plane of Order 8.- 37. Soluble Flag Transitive Collineation Groups.- 38. Some Characterizations of Finite Desarguesian Planes.- 39. Translation Planes Whose Collineation Group Acts Doubly Transitively on l?.- 40. A Theorem of Burmester and Hughes.- 41. Bol Planes.- VII Translation Planes of Order q2 Admitting SL(2, q) as a Collineation Group.- 42. Ovals in Finite Desarguesian Planes.- 43. Twisted Cubics.- 44. Irreducible Representations of SL(2,2r).- 45. The Hering and the Schäffer Planes.- 46. Three Planes of Order 25.- 47. Quasitransvections.- 48. Desarguesian Spreads in V(4, q).- 49. Translation Planes of Order q2 Admitting SL(2, q) as a Collineation Group.- 50. The Collineation Groups of the Hering and Schäffer Planes.- 51. The Theorem of Cofman-Prohaska.- 52. Prohaska’s Characterization of the Hall Planes.- Index of Special Symbols.