## 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

An Overview | 1 |

Partial Spreads and Translation Nets | 13 |

Quasifields | 29 |

Copyright | |

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### Common terms and phrases

2-dimensional Abelian acts transitively admits a collineation affine homology groups associated translation plane Assume automorphism axis Baer group Baer subplanes bijection collineation group components conical flock consider constructed contains coordinatized Corollary corresponding cyclic group cyclic homology defined Definition denote Desarguesian affine plane Desarguesian plane Desarguesian spread doubly transitive elation group finite fixed flag-transitive Foulser Furthermore GF(q GF(qn GF{q group G group isomorphic group of order Hence intersection isomorphic Jha and Johnson kernel homology group Klein quadric Lemma lifted line at infinity line spreads linear maps matrix nearfield plane non-Desarguesian normal subgroup odd order orbits of length order q2 ovoid partial spread plane of order plane with spread points projective space quadratic cone quasifield replacement semifield plane semifield spread spread in PG(3 spread of order spread set subfield subgeometry partitions subgroup subregular planes subregular spreads subset subspaces Theorem translation complement translation plane twisted field plane vector space