Handbook of Finite Translation Planes

Handbook of Finite Translation Planes

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The Handbook of Finite Translation Planes provides a comprehensive listing of all translation planes derived from a fundamental construction technique, an explanation of the classes of translation planes using both descriptions and construction methods, and thorough sketches of the major relevant theorems. From the methods of Andre to coordinate and linear algebra, the book unifies the numerous diverse approaches for analyzing finite translation planes. It pays particular attention to the processes that are used to study translation planes, including ovoid and Klein quadric projection, multiple derivation, hyper-regulus replacement, subregular lifting, conical distortion, and Hermitian sequences. In addition, the book demonstrates how the collineation group can affect the structure of the plane and what information can be obtained by imposing group theoretic conditions on the plane. The authors also examine semifield and division ring planes and introduce the geometries of two-dimensional translation planes. As a compendium of examples, processes, construction techniques, and models, the Handbook of Finite Translation Planes equips readers with precise information for finding a particular plane. It presents the classification results for translation planes and the general outlines of their proofs, offers a full review of all recognized construction techniques for translation planes, and illustrates known examples.show more

Product details

  • Hardback | 888 pages
  • 162.6 x 236.2 x 50.8mm | 1,338.11g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 1584886056
  • 9781584886051

About Mauro Biliotti

University of Iowa, Iowa City, IA, USA Caledonian University, Glasgow, Scotland University Di Leece/Matematica, Leece, Italyshow more

Table of contents

Preface and Acknowledgments An Overview Translation Plane Structure Theory Partial Spreads and Translation Nets Partial Spreads and Generalizations Quasifields Derivation Frequently Used Tools Sharply Transitive Sets SL(2, p) x SL(2, p)-Planes Classical Semifields Groups of Generalized Twisted Field Planes Nuclear Fusion in Semifields Cyclic Semifields T-Cyclic GL(2, q)-Spreads Cone Representation Theory Andre Net Replacements and Ostrom-Wilke Generalizations Foulser's ?-Planes Regulus Lifts, Intersections over Extension Fields Hyper-Reguli Arising from Andre Hyper-Reguli Translation Planes with Large Homology Groups Derived Generalized Andre Planes The Classes of Generalized Andre Planes C-System Nearfields Subregular Spreads Fano Configurations Fano Configurations in Generalized Andre Planes Planes with Many Elation Axes Klein Quadric Parallelisms Transitive Parallelisms Ovoids Known Ovoids Simple T-Extensions of Derivable Nets Baer Groups on Parabolic Spreads Algebraic Lifting Semifield Planes of Orders q4, q6 Known Classes of Semifields Methods of Oyama and the Planes of Suetake Coupled Planes Hyper-Reguli Subgeometry Partitions Groups on Multiple Hyper-Reguli Hyper-Reguli of Dimension 3 Elation-Baer Incompatibility Hering-Ostrom Elation Theorem Baer-Elation Theory Spreads Admitting Unimodular Sections-Foulser-Johnson Theorem Spreads of Order q2-Groups of Order q2 Transversal Extensions Indicator Sets Geometries and Partitions Maximal Partial Spreads Sperner Spaces Conical Flocks Ostrom and Flock Derivation Transitive Skeletons BLT-Set Examples Many Ostrom-Derivates Infinite Classes of Flocks Sporadic Flocks Hyperbolic Fibrations Spreads with Many Homologies Nests of Reguli Chains Multiple Nests A Few Remarks on Isomorphisms Flag-Transitive Geometries Quartic Groups in Translation Planes Double Transitivity Triangle Transitive Planes Hiramine-Johnson-Draayer Theory Bol Planes 2/3-Transitive Axial Groups Doubly Transitive Ovals and Unitals Rank 3 Affine Planes Transitive Extensions Higher-Dimensional Flocks j...j-Planes Orthogonal Spreads Symplectic Groups-The Basics Symplectic Flag-Transitive Spreads Symplectic Spreads When Is a Spread Not Symplectic? When Is a Spread Symplectic? The Translation Dual of a Semifield Unitals in Translation Planes Hyperbolic Unital Groups Transitive Parabolic Groups Doubly Transitive Hyperbolic Unital Groups Retraction Multiple Spread Retraction Transitive Baer Subgeometry Partitions Geometric and Algebraic Lifting Quasi-Subgeometry Partitions Hyper-Regulus Partitions Small-Order Translation Planes Dual Translation Planes and Their Derivates Affine Planes with Transitive Groups Cartesian Group Planes-Coulter-Matthews Planes Admitting PGL(3, q) Planes of Order = 25 Real Orthogonal Groups and Lattices Aspects of Symplectic and Orthogonal Geometry Fundamental Results on Groups Atlas of Planes and Processes Bibliography Theorems Models General Indexshow more