Handbook of Finite Translation Planes

Norman Johnson, Vikram Jha, Mauro Biliotti9781584886051, 1584886056

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”

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Table of contents :
Title……Page 2
Copyright……Page 5
Contents……Page 8
Preface and Acknowledgments……Page 20
CHAPTER 1: An Overview……Page 24
CHAPTER 2: Translation Plane Structure Theory……Page 28
CHAPTER 3: Partial Spreads and Translation Nets……Page 36
CHAPTER 4: Partial Spreads and Generalizations……Page 40
CHAPTER 5: Quasifields……Page 52
CHAPTER 6: Derivation……Page 70
CHAPTER 7: Frequently Used Tools……Page 78
CHAPTER 8: Sharply Transitive Sets……Page 82
CHAPTER 9: SL(2, p) × SL(2, p)-Planes……Page 86
CHAPTER 10: Classical Semifields……Page 94
CHAPTER 11: Groups of Generalized Twisted Field Planes……Page 100
CHAPTER 12: Nuclear Fusion in Semifields……Page 114
CHAPTER 13: Cyclic Semifields……Page 132
CHAPTER 14: T-Cyclic GL(2, q)-Spreads……Page 136
CHAPTER 15: Cone Representation Theory……Page 140
CHAPTER 16: Andre Net Replacements and Ostrom–Wilke Generalizations……Page 146
CHAPTER 17: Foulser’s Lambda-Planes……Page 154
CHAPTER 18: Regulus Lifts, Intersections over Extension Fields……Page 166
CHAPTER 19: Hyper-Reguli Arising from Andre Hyper-Reguli……Page 176
CHAPTER 20: Translation Planes with Large Homology Groups……Page 182
CHAPTER 21: Derived Generalized Andre Planes……Page 188
CHAPTER 22: The Classes of Generalized Andre Planes……Page 192
CHAPTER 23: C-System Nearfields……Page 194
CHAPTER 24: Subregular Spreads……Page 198
CHAPTER 25: Fano Configurations……Page 234
CHAPTER 26: Fano Configurations in Generalized Andre Planes……Page 240
CHAPTER 27: Planes with Many Elation Axes……Page 242
CHAPTER 28: Klein Quadric……Page 246
CHAPTER 29: Parallelisms……Page 248
CHAPTER 30: Transitive Parallelisms……Page 258
CHAPTER 31: Ovoids……Page 264
CHAPTER 32: Known Ovoids……Page 268
CHAPTER 33: Simple T-Extensions of Derivable Nets……Page 272
CHAPTER 34: Baer Groups on Parabolic Spreads……Page 280
CHAPTER 35: Algebraic Lifting……Page 286
CHAPTER 36: Semifield Planes of Orders q4, q6……Page 294
CHAPTER 37: Known Classes of Semifields……Page 300
CHAPTER 38: Methods of Oyama, and the Planes of Suetake……Page 306
CHAPTER 39: Coupled Planes……Page 312
CHAPTER 40: Hyper-Reguli……Page 320
CHAPTER 41: Subgeometry Partitions……Page 330
CHAPTER 42: Groups on Multiple Hyper-Reguli……Page 334
CHAPTER 43: Hyper-Reguli of Dimension 3……Page 338
CHAPTER 44: Elation-Baer Incompatibility……Page 346
CHAPTER 45: Hering–Ostrom Elation Theorem……Page 352
CHAPTER 46: Baer Elation Theory……Page 356
CHAPTER 47: Spreads Admitting Unimodular Sections—Foulser–Johnson Theorem……Page 360
CHAPTER 48: Spreads of Order q2—Groups of Order q2……Page 374
CHAPTER 49: Transversal Extensions……Page 380
CHAPTER 50: Indicator Sets……Page 396
CHAPTER 51: Geometries and Partitions……Page 416
CHAPTER 52: Maximal Partial Spreads……Page 428
CHAPTER 53: Sperner Spaces……Page 430
CHAPTER 54: Conical Flocks……Page 432
CHAPTER 55: Ostrom and Flock Derivation……Page 444
CHAPTER 56: Transitive Skeletons……Page 454
CHAPTER 57: BLT-Set Examples……Page 456
CHAPTER 58: Many Ostrom-Derivates……Page 460
CHAPTER 59: Infinite Classes of Flocks……Page 462
CHAPTER 60: Sporadic Flocks……Page 468
CHAPTER 61: Hyperbolic Fibrations……Page 472
CHAPTER 62: Spreads with `Many’ Homologies……Page 484
CHAPTER 63: Nests of Reguli……Page 494
CHAPTER 64: Chains……Page 508
CHAPTER 65: Multiple Nests……Page 514
CHAPTER 66: A Few Remarks on Isomorphisms……Page 524
CHAPTER 67: Flag-Transitive Geometries……Page 526
CHAPTER 68: Quartic Groups in Translation Planes……Page 532
CHAPTER 69: Double Transitivity……Page 538
CHAPTER 70: Triangle Transitive Planes……Page 544
CHAPTER 71: Hiramine–Johnson–Draayer Theory……Page 546
CHAPTER 72: Bol Planes……Page 552
CHAPTER 73: 2/3-Transitive Axial Groups……Page 554
CHAPTER 74: Doubly Transitive Ovals and Unitals……Page 558
CHAPTER 75: Rank 3 Affine Planes……Page 562
CHAPTER 76: Transitive Extensions……Page 566
CHAPTER 77: Higher-Dimensional Flocks……Page 578
CHAPTER 78: j…j-Planes……Page 592
CHAPTER 79: Orthogonal Spreads……Page 606
CHAPTER 80: Symplectic Groups—The Basics……Page 612
CHAPTER 81: Symplectic Flag-Transitive Spreads……Page 620
CHAPTER 82: Symplectic Spreads……Page 642
CHAPTER 83: When Is a Spread Not Symplectic?……Page 654
CHAPTER 84: When Is a Spread Symplectic?……Page 664
CHAPTER 85: The Translation Dual of a Semifield……Page 666
CHAPTER 86: Unitals in Translation Planes……Page 672
CHAPTER 87: Hyperbolic Unital Groups……Page 684
CHAPTER 88: Transitive Parabolic Groups……Page 694
CHAPTER 89: Doubly Transitive Hyperbolic Unital Groups……Page 696
CHAPTER 90: Retraction……Page 700
CHAPTER 91: Multiple Spread Retraction……Page 716
CHAPTER 92: Transitive Baer Subgeometry Partitions……Page 724
CHAPTER 93: Geometric and Algebraic Lifting……Page 732
CHAPTER 94: Quasi-Subgeometry Partitions……Page 738
CHAPTER 95: Hyper-Regulus Partitions……Page 754
CHAPTER 96: Small-Order Translation Planes……Page 760
CHAPTER 97: Dual Translation Planes and Their Derivates……Page 768
CHAPTER 98: Affine Planes with Transitive Groups……Page 772
CHAPTER 99: Cartesian Group Planes—Coulter–Matthews……Page 774
CHAPTER 100: Planes Admitting PGL(3, q)……Page 778
CHAPTER 101: Planes of Order………Page 780
CHAPTER 102: Real Orthogonal Groups and Lattices……Page 782
CHAPTER 103: Aspects of Symplectic and Orthogonal Geometry……Page 786
CHAPTER 104: Fundamental Results on Groups……Page 804
CHAPTER 105: Atlas of Planes and Processes……Page 812
Bibliography……Page 830
Theorems……Page 872
Models……Page 876
General Index……Page 880