Handbook of Finite Translation Planes

Completing a three-volume work by the authors, this handbook provides a complete description of all known finite translation planes. It presents the classification results and proofs for translation planes, offers a full review of all recognized construction techniques for translation planes, and illustrates known examples. The authors describe derivable nets, parallelisms, translation nets and planes, spreads, classes, and various geometries intrinsic to translation planes, including flocks of quadric sets and generalized quadrangles. As a compendium of examples, processes, construction techniques, and models, the handbook equips you with precise information for finding a particular plane.

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.

Preface and AcknowledgmentsAn OverviewTranslation Plane Structure TheoryPartial Spreads and Translation NetsPartial Spreads and GeneralizationsQuasifieldsDerivationFrequently Used ToolsSharply Transitive SetsSL(2, p) × SL(2, p)-PlanesClassical SemifieldsGroups of Generalized Twisted Field PlanesNuclear Fusion in SemifieldsCyclic SemifieldsT-Cyclic GL(2, q)-SpreadsCone Representation TheoryAndré Net Replacements and Ostrom-Wilke GeneralizationsFoulser's ?-PlanesRegulus Lifts, Intersections over Extension FieldsHyper-Reguli Arising from André Hyper-ReguliTranslation Planes with Large Homology GroupsDerived Generalized André PlanesThe Classes of Generalized André PlanesC-System NearfieldsSubregular SpreadsFano ConfigurationsFano Configurations in Generalized André Planes Planes with Many Elation Axes Klein QuadricParallelismsTransitive ParallelismsOvoidsKnown OvoidsSimple T-Extensions of Derivable NetsBaer Groups on Parabolic SpreadsAlgebraic LiftingSemifield Planes of Orders q4, q6Known Classes of SemifieldsMethods of Oyama and the Planes of Suetake Coupled Planes Hyper-Reguli Subgeometry PartitionsGroups on Multiple Hyper-Reguli Hyper-Reguli of Dimension 3Elation-Baer IncompatibilityHering-Ostrom Elation Theorem Baer-Elation TheorySpreads Admitting Unimodular Sections-Foulser-Johnson TheoremSpreads of Order q2-Groups of Order q2Transversal ExtensionsIndicator SetsGeometries and Partitions Maximal Partial Spreads Sperner Spaces Conical FlocksOstrom and Flock Derivation Transitive Skeletons BLT-Set ExamplesMany Ostrom-Derivates Infinite Classes of FlocksSporadic FlocksHyperbolic FibrationsSpreads with Many HomologiesNests of ReguliChains Multiple NestsA Few Remarks on Isomorphisms Flag-Transitive GeometriesQuartic Groups in Translation PlanesDouble TransitivityTriangle Transitive PlanesHiramine-Johnson-Draayer TheoryBol Planes 2/3-Transitive Axial GroupsDoubly Transitive Ovals and UnitalsRank 3 Affine Planes Transitive ExtensionsHigher-Dimensional Flocks j…j-PlanesOrthogonal Spreads Symplectic Groups-The BasicsSymplectic Flag-Transitive Spreads Symplectic Spreads When Is a Spread Not Symplectic? When Is a Spread Symplectic?The Translation Dual of a SemifieldUnitals in Translation PlanesHyperbolic Unital GroupsTransitive Parabolic Groups Doubly Transitive Hyperbolic Unital GroupsRetractionMultiple Spread RetractionTransitive Baer Subgeometry PartitionsGeometric and Algebraic Lifting Quasi-Subgeometry PartitionsHyper-Regulus Partitions Small-Order Translation PlanesDual Translation Planes and Their Derivates Affine Planes with Transitive Groups Cartesian Group Planes-Coulter-Matthews Planes Admitting PGL(3, q) Planes of Order = 25Real Orthogonal Groups and Lattices Aspects of Symplectic and Orthogonal Geometry Fundamental Results on Groups Atlas of Planes and ProcessesBibliography Theorems Models General Index