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" Translation generalized quadrangles / "
J.A. Thas, K. Thas, H. Van Maldeghem.
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BL
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| Record Number
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1004502
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b758872
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| Main Entry
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Thas, J. A., (Joseph Adolf)
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| Title & Author
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Translation generalized quadrangles /\ J.A. Thas, K. Thas, H. Van Maldeghem.
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| Publication Statement
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Singapore ;London :: World Scientific,, ©2006.
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| Series Statement
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Series in pure mathematics ;; v. 26
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1 online resource (xxx, 345 pages)
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| ISBN
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9789812772916
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: 981277291X
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9789812569516
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9812569510
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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1. Generalized quadrangles. 1.1. Finite generalized quadrangles. 1.2. Automorphisms. 1.3. Grids and dual grids. 1.4. The classical generalized quadrangles. 1.5. The generalized quadrangles T[symbol](O) and T[symbol](C) of tits. 1.6. The generalized quadrangles T[symbol](O). 1.7. Orders of the known generalized quadrangles. 1.8. Generalized quadrangles with small parameters -- 2. Regularity, antiregularity and 3-regularity. 2.1. Regularity. 2.2. Regularity and dual nets. 2.3. Antiregularity. 2.4. Antiregularity and Laguerre planes. 2.5. 3-regularity. 2.6. 3-regularity and subquadrangles. 2.7. 3-regularity, inversive planes, and characterizations -- 3. Elation and translation generalized quadrangles. 3.1. Some notions from group theory. 3.2. Elation generalized quadrangles. 3.3. Translation generalized quadrangles. 3.4. The kernel of a translation generalized quadrangle. 3.5. T(n, m, q)s and translation generalized quadrangles. 3.6. Regular pseudo-ovals and regular pseudo-ovoids. 3.7. Automorphisms of translation generalized quadrangles. 3.8. Important properties of O(n, m, q). 3.9. Pseudo-ovals. 3.10. Eggs. 3.11. The stabilizer of the base-point of a translation generalized quadrangle. 3.12. Structure of the automorphism group of a translation quadrangle -- 4. Generalized quadrangles and flocks. 4.1. Flocks. 4.2. Flocks and translation planes. 4.3. Flocks of ovoids and hyperbolic quadrics. 4.4. Flocks of cones. 4.5. Semifield flocks. Known examples of semifield flocks. 4.6. Generalized quadrangles and flocks. 4.7. Semifield flocks and translation generalized quadrangles. 4.8. Derivation and BLT-sets. 4.9. Constructions. 4.10. Property (G) for generalized quadrangles of order (s, s[symbol]). 4.11. Flocks, subquadrangles and ovals. Addendum A: isomorphisms of flock quadrangles and associated geometries. 4.12. The fundamental theorem of q-clan geometry, and applications. Addendum B: Basic questions on elation groups. 4.13. The standard conjectures and questions. 4.14. Some results by Payne and K. Thas. 4.15. Elation generalized quadrangles with nonisomorphic elation groups -- 5. Good eggs. 5.1. Good eggs and good translation generalized quadrangles. 5.2. Good eggs and veronese surfaces. 5.3. Coordinatization and applications -- 6. Generalized quadrangles, nets and the axiom of Veblen. 6.1. Generalized quadrangles and the axiom of Veblen. 6.2. Translation generalized quadrangles and the axiom of Veblen. 6.3. Property (G) and the axiom of Veblen. 6.4. Flock generalized quadrangles and the axiom of Veblen. 6.5. Subquadrangles and the axiom of Veblen. 6.6. Nets and characterizations of translation generalized quadrangles -- 7. Ovoids and subquadrangles. 7.1. Ovoids of Q(4, q). 7.2. Subquadrangles and ovoids. 7.3. Translation ovoids and semifield flocks. 7.4. Coordinates of the known nonclassical ovoids of Q(4, q). 7.5. Subquadrangles of T(O), with O good: the even case. 7.6. Subquadrangles of T(O), with O good: the odd case. 7.7. Subquadrangles: remaining cases and some applications. 7.8. Translation generalized quadrangles with one classical subquadrangle. 7.9. Elation generalized quadrangles with a subquadrangle -- 8. Translation generalized ovals. 8.1. Translation generalized ovoids and translation generalized ovals. 8.2. Note on the definition of translation generalized oval/ovoid. 8.3. Characterizations of the T2 (O) of tits. 8.4. A characterization of translation generalized ovals. 8.5. Classification of 2-transitive generalized ovals in even characteristic -- 9. Moufang sets and translation Moufang sets. 9.1. Definition and general results. 9.2. Finite Moufang sets -- 10. Configurations of translation points. 10.1. Span-symmetric generalized quadrangles. 10.2. Groups admitting a 4-Gonal basis. 10.3. SPGQs and Moufang sets. 10.4. Basic structural Lemmas. 10.5. Classification of SPGQs of order (s, t), 1<s[symbol]t<s[symbol]. 10.6. SPGQs of order (s, s[symbol]). 10.7. Generalized quadrangles with a line of translation points. 10.8. On the classification of translation generalized quadrangles -- 11. Moufang quadrangles with a translation point. 11.1. Notation. 11.2. Some general elementary Lemmas. 11.3. The Moufang property and analogues. 11.4. Tits generalized quadrangles and tits systems. 11.5. Properties of Moufang quadrangles. 11.6. Half 3-Moufang quadrangles. 11.7. 2-Moufang quadrangles and Fong-Seitz quadrangles. 11.8. Conclusion -- 12. Translation ovoids in translation quadrangles. 12.1. Ovoids, elation or translation with respect to a flag or a point. 12.2. Self-polar elation generalized quadrangles. 12.3. Suzuki-Tits Moufang sets. 12.4. Subtended elation ovoids -- 13. Translation generalized quadrangles in projective space. 13.1. Generalities about Lax embeddings. 13.2. Planar translation-homogeneous embeddings. 13.3. Exceptional non-planar translation-homogeneous embeddings. 13.4. Non-planarly embedded small translation generalized quadrangles. 13.5. Non-planarly embedded translation generalized quadrangles.
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| Abstract
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Translation generalized quadrangles play a key role in the theory of generalized quadrangles, comparable to the role of translation planes in the theory of projective and affine planes. The notion of translation generalized quadrangle is a local analogue of the more global "Moufang Condition", a topic of great interest, also due to the classification of all Moufang polygons. Attention is thus paid to recent results in that direction, but also many of the most important results in the general theory of generalized quadrangles that appeared since 1984 are treated. Translation Generalized Quadran.
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| Subject
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Finite generalized quadrangles.
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| Subject
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Finite generalized quadrangles.
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| Subject
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MATHEMATICS-- Geometry-- General.
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| Dewey Classification
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516.13
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| LC Classification
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QA167.2.T42 2006eb
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| Added Entry
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Maldeghem, H. van.
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Thas, Koen,1977-
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