Handbook of Finite Translation Planes  book cover
1st Edition

Handbook of Finite Translation Planes

ISBN 9781584886051
Published February 15, 2007 by Chapman and Hall/CRC
888 Pages

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Book Description

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
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
Transitive Parallelisms
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
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
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
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
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
General Index

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