Handbook of Finite Translation Planes (Pure and Applied Mathematics)

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