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.
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.
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-Suetake
Planes. 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.
Norman Johnson, Vikram Jha, Mauro Biliotti
"The authors, who are the undisputed leaders in the subject, present the huge material in shorter but virtually independent chapters, each dedicated to a particular aspect, such as the connection between translation planes and quasifields... This book highly recommended for the very clear, rigorous and detailed expostion and cannot be missing in the library of any researcher in Geometry."-Bambina Larato, Zentralblatt MATH, 2008, 1136
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