## Handbook of Finite Translation PlanesThe 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

1 | |

5 | |

13 | |

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 |

849 | |

853 | |

857 | |

### Common terms and phrases

2-dimensional Abelian acts transitively admits a collineation affine homology groups André nets André planes associated translation plane axis Baer group Baer subplanes bijection collineation group components consider constructed contains coordinatized Corollary corresponding cyclic cyclic homology defined Definition denote derivable Desarguesian affine plane Desarguesian plane Desarguesian spread doubly transitive elation group fibration fixes flock spread Foulser Furthermore Geom GF(q group G group isomorphic group of order Hence hyper-reguli intersection Jha and Johnson kernel homology group Lemma line at infinity linear maps Math matrix mutually disjoint nearfield plane odd order orbits of length order q orthogonal ovoid partial spread plane of order plane with spread points projective space quadratic cone quadratic form quasifield regulus semifield plane semifield spread spread in PG(3,q spread set subgeometry partition subgroup subregular planes subspaces symplectic spread Theorem translation complement translation plane twisted field planes vector space