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.     

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     

Compact Ovoids in Quadrangles I: Geometric Constructions

This paper is about ovoids in infinite generalized quadrangles. Using the axiom of choice, Cameron showed that infinite quadrangles contain many ovoids. Therefore, we consider mainly closed ovoids in compact quadrangles. After deriving some basic properties of compact ovoids, we consider ovoids which arise from full imbeddings. This leads to restrictions for the topological parameters (m,m). For example, if there is a regular pair of lines or a full closed subquadrangle, then mm. The existence of full subquadrangles implies the nonexistence of ideal subquadrangles, so finite-dimensional quadrangles are either point-minimal or line-minimal. Another result is that (up to duality) such a quadrangle is spanned by the set of points on an ordinary quadrangle. This is useful for studying orbits of automorphism groups. Finally we prove general nonexistence results for ovoids in quadrangles with low-dimensional line pencils. As one consequence we show that the symplectic quadrangle over an algebraically closed field of characteristic 0 has no Zariski-closed ovoids or spreads.     

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.      

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.     

Notes on elation generalized quadrangles

Let be a finite generalized quadrangle of order (s,t),s,t>1. An “elation about a point p” of is an automorphism fixing p linewise and fixing no point which is not collinear with p. An elation that generates a cyclic group of elations is called a “standard elation”. One of the problems already considered in Payne and Thas (Finite Generalized Quadrangles (1984)) is to determine just when the set of elations about the point (∞) is a group. The purpose of this paper is to provide an example where this is not the case, and then to show that for a flock generalized quadrangle the usual group of elations about (∞) is the complete set of standard elations about (∞).     

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.     

Independence of axioms for fourgonal families

We show, by means of (counter)examples, that the axioms for fourgonal families (as used to construct elation generalized quadrangles) are independent. Received 22 September 2000.     

A Theorem Concerning Nets Arising from Generalized Quadrangles with a Regular Point

Suppose is a generalized quadrangle (GQ) of order , with a regular point. Then there is a net which arises from this regular point. We prove that if such a net has a proper subnet with the same degree as the net, then it must be an affine plane of order t. Also, this affine plane induces a proper subquadrangle of order t containing the regular point, and we necessarily have that . This result has many applications, of which we give one example. Suppose is an elation generalized quadrangle (EGQ) of order , with elation point p. Then is called a skew translation generalized quadrangle (STGQ) with base-point p if there is a full group of symmetries about p of order t which is contained in the elation group. We show that a GQ of order s is an STGQ with base-point p if and only if p is an elation point which is regular.     

Generalized quadrangles admitting a sharply transitive Heisenberg group

All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.      

Compact Generalised Quadrangles with Parameter 1 and Large Group of Automorphisms

If the group of automorphisms of a compact generalised quadrangle with parameter 1 has dimension at least 6, the quadrangle is the real symplectic quadrangle or its dual. There are nonclassical compact generalised quadrangles with parameter 1 whose group of automorphism has dimension 5.     

Affine ovoids and extensions of generalized quadrangles

A set Δ of vertices of a generalized quadrangle of order (s, t) is said to be a hyperoval if any line intersects Δ in either 0, or 2 points. A hyperoval Δ is called an affine ovoid if |Δ|=2st. It is well known that μ-subgraphs in triangular extensions of generalized quadrangles are hyperovals. In the present paper we prove that ifS is a triangular extension forGQ(s, t) with totally regular point graph Γ such that μ=2st, thens is even, Γ is an τ-antipodal graph of diameter 3 with τ=1+s/2, and eithers=2, ort=s+2. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 266–271, August, 2000.     

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.     

On the nonexistence of certain Hughes generalized quadrangles

In this paper, we prove that the Hermitian quadrangle is the unique generalized quadrangle Γ of order (q ², q ³) containing some subquadrangle of order (q ², q) isomorphic to such that every central elation of the subquadrangle is induced by a collineation of Γ. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

Characterizations of Translation Generalized Quadrangles

If x is a regular point of the generalizedquadrangle of order (s,t), s 1 t, then x defines a dual net . If contains a line L of regularpoints and if for at least one point x on Lthe automorphism group of the dual net satisfies certain transitivityproperties, then is a translation generalized quadrangle. Thisresult has many applications. We give one example. Ifs=t 1, then is a dual affine plane. Let be a generalizedquadrangle of orders,s odd and s 1, which contains a lineL of regular points. If for at least one pointx on L the plane is Desarguesian, then is isomorphic to the classical generalizedquadrangleW(s).     

Semi-Biplanes and Antiregular Generalied Quadrangles

A natural method to construct semi-biplanes from antiregular generalized quadrangles is introduced. Properties of the semi-biplanes constructed are discussed. In the finite case and in the topological case the semi-biplanes that arise bear a strong resemblance to semi-biplanes that arise in the natural way from projective planes admitting an involutory homology.     

Transitive affine planes admitting elations

Affine planes which admit a point transitive collineation group and at least one affine elation are considered. Such a plane is shown to be (A,?_∞)-transitive for some point A on ?_t8 and to be a translation plane if at least two distinct elation centers exist. If the plane has at least (order)^1/2+1 distinct elation centers and the group generated by the elations is nonsolvable then the plane is either Desarguesian or Lüneburg-Tits.     

A Sufficient Criterion for Homotopy Cartesianess

Suppose given a commutative quadrangle in a Verdier triangulated category such that there exists an induced isomorphism on the horizontally taken cones. Suppose that the endomorphism ring of the initial or the terminal corner object of this quadrangle satisfies a finiteness condition. Then this quadrangle is homotopy cartesian.     

Moufang sets from groups of mixed type

We consider the existence of Moufang sets related to certain groups of mixed type. This way, we obtain new examples of Moufang sets and new constructions of known classes. The most interesting class of new examples is related to the Moufang quadrangles of type and to the Ree–Tits octagon over a nonperfect field, and the root groups of each member have nilpotency class three.