Handbook of Finite Translation Planes: 1st Edition (Hardback) book cover

Handbook of Finite Translation Planes

1st Edition

By Norman Johnson, Vikram Jha, Mauro Biliotti

Chapman and Hall/CRC

888 pages

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Hardback: 9781584886051
pub: 2007-02-15
eBook (VitalSource) : 9780429140280
pub: 2007-02-15
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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 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

About the Series

Chapman & Hall/CRC Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Geometry / General
MATHEMATICS / Combinatorics