Translation PlanesWir 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. |
Contents
Rank3Planes with an Orbit of Length 2 on the Line | 17 |
Chapter II | 25 |
Generalized André Planes | 51 |
Copyright | |
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2-subgroup of G acts transitively affine plane affine point André plane André system assume automorphism axis B₁ b₂ Baer subplane centre collineation group conjugate contradiction Corollary cyclic subgroup defined denote desarguesian plane Dickson nearfield distinct points divides q division ring elements of order exists exactly fixed point follows Furthermore G₁ GF(q GL(V group of order implies induces infer involution involutory involutory homology isomorphic kernel left invariant Lemma length q Let G Let Q Lüneburg mapping Maschke's theorem Moreover nearfield plane normal subgroup obtain operates irreducibly operates transitively orbit of length order q p-group P₁ plane of order point orbit projective plane PROOF proves quasifield quaternion group Schur's lemma set of points space of rank subgroup of order subspaces of rank Sylow 2-subgroup tangent Theorem translation plane vector space whence yields