عنوان

Translation generalized quadrangles /

(Number (ISBN

9789812772916

(Number (ISBN

981277291X

Erroneous ISBN

9789812569516

Erroneous ISBN

9812569510

Number

b758872

Title Proper

Translation generalized quadrangles /

General Material Designation

[Book]

First Statement of Responsibility

J.A. Thas, K. Thas, H. Van Maldeghem.

Place of Publication, Distribution, etc.

London :

Name of Publisher, Distributor, etc.

World Scientific,

Date of Publication, Distribution, etc.

©2006.

Specific Material Designation and Extent of Item

1 online resource (xxx, 345 pages)

Series Title

Series in pure mathematics ;

Volume Designation

v. 26

Text of Note

Includes bibliographical references and index.

Text of Note

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

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