Handbook of Finite Translation Planes (Google eBook)

Front Cover
CRC Press, Feb 15, 2007 - Mathematics - 888 pages
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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.
  

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Contents

An Overview
1
Translation Plane Structure Theory
5
Partial Spreads and Translation Nets
13
Partial Spreads and Generalizations
17
Quasifields
29
Derivation
47
Frequently Used Tools
55
Sharply Transitive Sets
59
Transitive Skeletons
431
BLTSet Examples
433
Many OstromDerivates
437
Infinite Classes of Flocks
439
Sporadic Flocks
445
Hyperbolic Fibrations
449
Spreads with Many Homologies
461
Nests of Reguli
471

SL2 p SL2 pPlanes
63
Classical Semifields
71
Groups of Generalized Twisted Field Planes
77
Nuclear Fusion in Semifields
91
Cyclic Semifields
109
TCyclic GL2 qSpreads
113
Cone Representation Theory
117
Andre Net Replacements and OstromWilke Generalizations
123
Foulsers LambdaPlanes
131
Regulus Lifts Intersections over Extension Fields
143
HyperReguli Arising from Andre HyperReguli
153
Translation Planes with Large Homology Groups
159
Derived Generalized Andre Planes
165
The Classes of Generalized Andre Planes
169
CSystem Nearfields
171
Subregular Spreads
175
Fano Configurations
211
Fano Configurations in Generalized Andre Planes
217
Planes with Many Elation Axes
219
Klein Quadric
223
Parallelisms
225
Transitive Parallelisms
235
Ovoids
241
Known Ovoids
245
Simple TExtensions of Derivable Nets
249
Baer Groups on Parabolic Spreads
257
Algebraic Lifting
263
Semifield Planes of Orders q4 q6
271
Known Classes of Semifields
277
Methods of Oyama and the Planes of Suetake
283
Coupled Planes
289
HyperReguli
297
Subgeometry Partitions
307
Groups on Multiple HyperReguli
311
HyperReguli of Dimension 3
315
ElationBaer Incompatibility
323
HeringOstrom Elation Theorem
329
Baer Elation Theory
333
Spreads Admitting Unimodular SectionsFoulserJohnson Theorem
337
Spreads of Order q2Groups of Order q2
351
Transversal Extensions
357
Indicator Sets
373
Geometries and Partitions
393
Maximal Partial Spreads
405
Sperner Spaces
407
Conical Flocks
409
Ostrom and Flock Derivation
421
Chains
485
Multiple Nests
491
A Few Remarks on Isomorphisms
501
FlagTransitive Geometries
503
Quartic Groups in Translation Planes
509
Double Transitivity
515
Triangle Transitive Planes
521
HiramineJohnsonDraayer Theory
523
Bol Planes
529
23Transitive Axial Groups
531
Doubly Transitive Ovals and Unitals
535
Rank 3 Affine Planes
539
Transitive Extensions
543
HigherDimensional Flocks
555
jjPlanes
569
Orthogonal Spreads
583
Symplectic GroupsThe Basics
589
Symplectic FlagTransitive Spreads
597
Symplectic Spreads
619
When Is a Spread Not Symplectic?
631
When Is a Spread Symplectic?
641
The Translation Dual of a Semifield
643
Unitals in Translation Planes
649
Hyperbolic Unital Groups
661
Transitive Parabolic Groups
671
Doubly Transitive Hyperbolic Unital Groups
673
Retraction
677
Multiple Spread Retraction
693
Transitive Baer Subgeometry Partitions
701
Geometric and Algebraic Lifting
709
QuasiSubgeometry Partitions
715
HyperRegulus Partitions
731
SmallOrder Translation Planes
737
Dual Translation Planes and Their Derivates
745
Affine Planes with Transitive Groups
749
Cartesian Group PlanesCoulterMatthews
751
Planes Admitting PGL3 q
755
Planes of Order
757
Real Orthogonal Groups and Lattices
759
Aspects of Symplectic and Orthogonal Geometry
763
Fundamental Results on Groups
781
Atlas of Planes and Processes
789
Bibliography
807
Theorems
849
Models
853
General Index
857
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