Handbook of Finite Translation Planes

Handbook of Finite Translation Planes

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ISBN 9781584886051
Cat# C6056

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Features

  • Provides a complete description of all known finite translation planes
  • Includes examples of translation planes and related geometries
  • Connects translation planes with numerous areas of incidence geometry such as flocks of quadric sets, generalized quadrangles, and partitions
  • Supplies all of the varied construction techniques of translation planes
  • Shows the importance of various classes of translation planes by illustrating their place within certain fundamental classification schemes
  • Summary

    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 André 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.

    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) × 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
    André Net Replacements and Ostrom-Wilke Generalizations
    Foulser's ?-Planes
    Regulus Lifts, Intersections over Extension Fields
    Hyper-Reguli Arising from André Hyper-Reguli
    Translation Planes with Large Homology Groups
    Derived Generalized André Planes
    The Classes of Generalized André Planes
    C-System Nearfields
    Subregular Spreads
    Fano Configurations
    Fano Configurations in Generalized André 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 Index