Affine Generalized Quadrangles – An Axiomatization

The complement of a geometric hyperplane of a generalized quadrangle is called an affine generalized quadrangle. Since a geometric hyperplane of a generalized quadrangle is either an ovoid or the perp of a point or a subquadrangle, there are three quite different classes of affine generalized quadrangles. The article proposes seven axioms (AQ1)–(AQ7) characterizing affine generalized quadrangles as point-line geometries. Certain subsets of the seven Axioms together with certain conditions distinguish what kind of hyperplane complement is realized. By just (AQ1)–(AQ6), finite affine generalized quadrangles are characterized completely.     

A generalized quadrangle of order (s, t) with center of transitivity is an elation quadrangle if s ≤ t

We show that a generalized quadrangle of order (s, t) with a center of transitivity is an elation generalized quadrangle if s ≤ t. In order to obtain this result, we generalize Frohardt’s result on Kantor’s conjecture from elation quadrangles to the more general case of quadrangles with a center of transitivity.      

The automorphism group of Payne derived generalized quadrangles

We solve a long-standing open problem by proving that the automorphism group of any thick Payne derived generalized quadrangle with ambient quadrangle S a thick generalized quadrangle of order s, s?5 and odd, with a center of symmetry, is induced by the automorphism group of S.     

Coordinate system and combinatorial objects (an example of a generalized quadrangle)

By using a generalized quadrangle as an example, we verify the assumption that coordinate systems (different from the standard Cartesian coordinate system) exist not only in an arbitrary projective plane, where they are determined by a nondegenerate quadrangle, but also in some other combinatorial objects.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 516–523, May, 1994.     

Linearly small elation quadrangles

We show that each elation generalized quadrangle with parameters (p, p), where p is a prime, is isomorphic to the symplectic quadrangle W(p) or its dual Q(4, p). Our results cover the more general case of linearly small elation generalized quadrangles. In particular, we obtain a characterization of the symplectic quadrangle over the field of complex numbers among compact connected quadrangles. We prove that every root elation quadrangle (Q, c, H _F ) is a skew translation quadrangle.     

Notes on binary codes related to the O(5, q) generalized quadrangle for odd q

In this note we determine the dimensions of the binary codes spanned by the lines or by the point neighborhoods in the generalized quadrangle Sp(4, q) and its dual O(5, q), where q is odd. Several more general results are given. As a side result we find that if a square generalized quadrangle of odd order has an antiregular point, then all of its points are antiregular.On leave from the Indian Statistical Institute, Calcutta; research supported by a grant from NWO.     

Central aspects of skew translation quadrangles, 1

Modulo a combination of duality, translation duality or Payne integration, every known finite generalized quadrangle except for the Hermitian quadrangles \(\mathcal {H}(4,q^2)\), is an elation generalized quadrangle for which the elation point is a center of symmetry—that is, is a “skew translation generalized quadrangle” (STGQ). In this series of papers, we classify and characterize STGQs. In the first installment of the series, (1) we obtain the rather surprising result that any skew translation quadrangle of finite odd order (s, s) is a symplectic quadrangle; (2) we determine all finite skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (3) we develop a structure theory for root elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root elations for each member, and hence, all members are “central” (the main property needed to control STGQs, as which will be shown throughout); and (4) we show that finite “generic STGQs,” a class of STGQs which generalizes the class of the previous item (but does not contain it by definition), have the expected parameters. We conjecture that the classes of (3) and (4) contain all STGQs.     

Spreads,flocks, and generalized quadrangles

A spread of PG(3,q), q an odd prime, recently constructed by R. Baker and G. Ebert, when generalized for q an odd prime power is isomorphic to a spread derived by J. A. Thas from a flock of a quadratic cone discovered by J. C. Fisher. The associated generalized quadrangle has an unusual colllneation.     

Domesticity in Generalized Quadrangles

An automorphism of a generalized quadrangle is called domestic if it maps no chamber, which is here an incident point-line pair, to an opposite chamber. We call it point-domestic if it maps no point to an opposite one and line-domestic if it maps no line to an opposite one. It is clear that a duality in a generalized quadrangle is always point-domestic and linedomestic. In this paper, we classify all domestic automorphisms of generalized quadrangles. Besides three exceptional cases occurring in the small quadrangles with orders (2, 2), (2, 4), and (3, 5), all domestic collineations are either point-domestic or line-domestic. Up to duality, they fall into one of three classes: Either they are central collineations, or they fix an ovoid, or they fix a large full subquadrangle. Remarkably, the three exceptional domestic collineatons in the small quadrangles mentioned above all have order 4.     

Generalized quadrangles of order 4. II

There is a unique generalized quadrangle of order 4.     

Three-dimensional quadrangles and flat Laguerre planes

Every three-dimensional generalized quadrangle can be constructed from flat Laguerre planes.Dedicated to Prof. H. Salzmann on his 60th birthday     

Connected Orbits in Topological Generalized Quadrangles

We study generalized quadrangles. After an investigation of the subgeometries that are generated by arbitrary sets of vertices, we consider orbits of connected subgroups of the automorphism group of topological generalized quadrangles. We deal with the problem of how a set of vertices has to be chosen in order that the union of the orbits generates a subquadrangle, or even the whole quadrangle.     

Half Moufang implies Moufang for finite generalized quadrangles

Summary A finite generalized quadrangle has two types of panels. If each panel of one type is Moufang, then every panel is Moufang. Hence by a theorem of Fong and Seitz [1] the quadrangle is classical or dual classical.Oblatum 1-XI-1989 & 7-XI-1990     

Generalized quadrangles constructed from topological Laguerre planes

The Lie geometry of a finite-dimensional locally compact connected Laguerre plane is a topological generalized quadrangle.     

Generalized quadrangles associated with G2(q)

If q ≡ 2 (mod 3), a generalized quadrangle with parameters q, q² is constructed from the generalized hexagon associated with the group G₂(q).     

Regular graphs constructed from the classical generalized quadrangle Q(4, q)

Let c_k be the smallest number of vertices in a regular graph with valency k and girth 8. It is known that c_{k + 1}?2(1 + k + k² + k³) with equality if and only if there exists a finite generalized quadrangle of order k. No such quadrangle is known when k is not a prime power. In this case, small regular graphs of valency k + 1 and girth 8 can be constructed from known generalized quadrangles of order q>k by removing a part of its structure. We investigate the case when q = k + 1 is a prime power, and try to determine the smallest graph under consideration that can be constructed from a generalized quadrangle of order q. This problem appears to be much more difficult than expected. We have general bounds and improve these for the classical generalized quadrangle Q(4, q), q even. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:70‐83, 2010     

On Finite Elation Generalized Quadrangles with Symmetries

We study the structure of finite groups G which act as elationgroups on finite generalized quadrangles and contain a fullgroup of symmetries about some line through the base point.Such groups are related to the translation groups of translationtransversal designs with parameters depending on those of thequadrangles. Using results on the structure of p-groups which act as translationgroups on transversal designs and results on the index of theHughes subgroups of finite p-groups, we can show how restrictedthe structure of elation groups of finite generalized quadrangleswith symmetries is. One of our main results is that G is necessarily an elementaryabelian 2-group, provided that G has even cardinality. In particular,the elation generalized quadrangle coordinatized by G is a translationgeneralized quadrangle with G as translation group, that is,G contains full groups of symmetries about every line throughthe base point.     

A geometric proof of a theorem on antiregularity of generalized quadrangles

A geometric proof is given in terms of Laguerre geometry of the theorem of Bagchi, Brouwer and Wilbrink, which states that if a generalized quadrangle of order s > 1 has an antiregular point then all of its points are antiregular.     

Structure of a code related to Sp(4, <Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">q</Emphasis> even

We determine the socle and the radical series of the binary code associated with a finite regular generalized quadrangle of even order, considered as a module for the commutator of each of the orthogonal subgroups in the corresponding symplectic group.     

All generalized quadrangles of order 3 are known

A generalized quadrangle of order 3 must be isomorphic either to the quadrangle $\text{P}$₄ or to its dual, where $\text{P}$₄ consists of all points of PG(3, 3) and those lines of PG(3, 3) self-conjugate with respect to a null polarity.