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QClan Geometries in Characteristic 2 (Frontiers in Mathematics)by Ilaria Cardinali
Synopses & ReviewsPublisher Comments:This monograph offers the only comprehensive, coherent treatment of the theory  in characteristic 2  of the socalled flock quadrangles, i.e., those generalized quadrangles (GQ) that arise from qclans, along with their associated ovals. Special attention is given to the determination of the complete oval stabilizers of each of the ovals associated with a flock GQ. A concise but logically complete introduction to the basic ideas is given. The theory of these flock GQ has evolved over the past two decades and has reached a level of maturation that makes it possible for the first time to give a satisfactory, unified treatment of all the known examples. The book will be a useful resource for all researchers working in the field of finite geometry, especially those interested in finite generalized quadrangles. It is of particular interest to those studying ovals in finite Desarguesian planes.
Synopsis:This book offers a complete proof of the Fundamental Theorem of qClan Geometry, followed by a detailed study of the known examples. It completely works out the collineation groups of the associated generalized quadrangles and the stabilizers of their associated ovals.
Synopsis:A qclan with q a power of 2 is equivalent to a certain generalized quadrangle with a family of subquadrangles each associated with an oval in the Desarguesian plane of order 2. It is also equivalent to a flock of a quadratic cone, and hence to a linespread of 3dimensional projective space and thus to a translation plane, and more. These geometric objects are tied together by the socalled Fundamental Theorem of qClan Geometry. The book gives a complete proof of this theorem, followed by a detailed study of the known examples. The collineation groups of the associated generalized quadrangles and the stabilizers of their associated ovals are worked out completely.
Table of ContentsIntroduction. 1. qClans and Their Geometries. 2. The Fundamental Theorem. 3. Aut(GQ(C)). 4. The Cyclic qClans. 5. Applications to the Known Cyclis qClans. 6. The Subiaco Oval Stabilizers. 7. The Adelaide Oval Stabilizers. 8. The Payne qClans. 9. Other Good Stuff. Bibliography.
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