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. |
Contents
An Overview | 1 |
Partial Spreads and Translation Nets | 13 |
Quasifields | 20 |
Copyright | |
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Common terms and phrases
2-dimensional acts transitively admits a collineation affine homology groups André nets André planes axis Baer group Baer subplanes bijection collineation group components consider constructed containing coordinatized COROLLARY corresponding cyclic cyclic homology defined DEFINITION denote derivable Desarguesian affine plane Desarguesian plane Desarguesian spread doubly transitive elation group fibration finite fixes flock spread Foulser Furthermore Geom GF q GF(q GF(q² 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 nearfield plane orbits of length order q orthogonal ovoid partial spread plane of order plane with spread points projective planes projective space quadratic cone quasifield regulus replacement semifield plane semifield spread spread in PG(3 spread set subgeometry partition subgroup subregular planes subspaces THEOREM translation complement translation plane twisted field planes vector space