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.     

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.      

Point regular groups of automorphisms of generalised quadrangles

We study the point regular groups of automorphisms of some of the known generalised quadrangles. In particular we determine all point regular groups of automorphisms of the thick classical generalised quadrangles. We also construct point regular groups of automorphisms of the generalised quadrangle of order (q−1,q+1) obtained by Payne derivation from the classical symplectic quadrangle W(3,q). For q=pf with f?2 we obtain at least two nonisomorphic groups when p?5 and at least three nonisomorphic groups when p=2 or 3. Our groups include nonabelian 2-groups, groups of exponent 9 and nonspecial p-groups. We also enumerate all point regular groups of automorphisms of some small generalised quadrangles.     

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.     

Slanted symplectic quadrangles

By slanting symplectic quadrangles W(F) over fieldsF, we obtain very simple examples of non-classical generalized quadrangles. We determine the collineation groups of these slanted quadrangles and their groups of projectivities. No slanted quadrangle is a topological quadrangle.     

The permutation action of finite symplectic groups of odd characteristic on their standard modules

We study the space of functions on a finite-dimensional vector space over a field of odd order as a module for a symplectic group. We construct a basis of this module with the following special properties. Each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule. From these properties, the dimension and composition factors of the submodule generated by any subset of the basis can be determined. These results apply to incidence geometry of the symplectic polar space, yielding the symplectic analogue of Hamada's additive formula for the p-ranks of the incidence matrices between points and flats. A special case leads to a closed formula for the p-rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle over a field of odd order. Together with earlier results on the 2-ranks, this result completes the determination of the p-ranks for these quadrangles.     

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.     

New constructions of two slim dense near hexagons

We provide a geometrical construction of the unique slim dense near hexagon with parameters (s,t,t₂)=(2,5,{1,2}). Using this construction, we construct the rank 3 symplectic dual polar space DSp(6,2) which is the unique slim dense near hexagon with parameters (s,t,t₂)=(2,6,2). Both near hexagons are constructed from two copies of the unique generalized quadrangle with parameters (2,2).     

Hemisystems of small flock generalized quadrangles

In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order (5², 5) and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order (s ², s) for s ≤ 11. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle H(3, s ²).     

Intriguing sets in partial quadrangles

The point‐line geometry known as a partial quadrangle (introduced by Cameron in 1975) has the property that for every point/line non‐incident pair (P, ?), there is at most one line through P concurrent with ?. So in particular, the well‐studied objects known as generalized quadrangles are each partial quadrangles. An intriguing set of a generalized quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalized quadrangles by Bamberg, Law and Penttila to partial quadrangles, which gives insight into the structure of hemisystems and other intriguing sets of generalized quadrangles. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:217‐245, 2011     

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 (∞).     

Determination of generalized quadrangles with distinct elation points

In this paper, we classify the finite generalized quadrangles of order (s,t), s,t > 1, which have a line L of elation points, with the additional property that there is a line M not meeting L for which {L, M} is regular. This is a first fundamental step towards the classification of those generalized quadrangles having a line of elation points. Mathematics Subject Classification (2000): 51E12, 51E20, 20B25, 20E42     

Symmetry in Generalized Quadrangles

In this paper, we describe some aspects of a Lenz(-Barlotti)-type classification of finite generalized quadrangles, which is being prepared by the author. Some new points of view are given. We also prove that each span-symmetric generalized quadrangle of order s > 1 with s even is isomorphic to $ \mathcal{Q} $ (4, s), without using the canonical connection (obtained by S. E. Payne in [15] between groups of order s ³ ? s with a 4-gonal basis and span-symmetric generalized quadrangle of order s. (The latter result was obtained for general s independently by W. M. Kantor in [10], and the author in [30] Finally, we obtain a classification program for all finite translation generalized quadrangles, which is suggested by the main results of [27], [30], [32], [35], [38] and [37].     

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.      

The uniqueness of 1-systems of W 5(q) satisfying the BLT-property, with q odd

In the symplectic polar space W ₅(q) every 1-system which satisfies the BLT-property (and then q is odd) defines a generalized quadrangle (GQ) of order (q ²,q ³). In this paper, we show that this 1-system is unique, so that the only GQ arising in this way is isomorphic to the classical GQ H(4,q ²), q odd.     

Local half Moufang quadrangles

In this article, we show that if every root of a finite generalized quadrangle containing a fixed point x is Moufang, then every dual root containing x in its interior is also Moufang. As a corollary, we obtain a new proof of the half Moufang theorem. This says that finite half Moufang quadrangles are Moufang.     

Compact Three-Dimensional Elation Quadrangles

We show that each compact three-dimensional EGQ is isomorphic either to the real symplectic quadrangle, or to a translation quadrangle of Tits type. In particular, the elation group is one of the two classical elation groups, and thus a simply connected nilpotent Lie group.     

Perfect dodecagon quadrangle systems

A dodecagon quadrangle is the graph consisting of two cycles: a 12-cycle (x₁,x₂,…,x₁₂) and a 4-cycle (x₁,x₄,x₇,x₁₀). A dodecagon quadrangle system of order n and index ρ [ DQS] is a pair (X,H), where X is a finite set of n vertices and H is a collection of edge disjoint dodecagon quadrangles (called blocks) which partitions the edge set of ρK_n, with vertex set X. A dodecagon quadrangle system of order n is said to be perfect [PDQS] if the collection of 4-cycles contained in the dodecagon quadrangles form a 4-cycle system of order n and index μ. In this paper we determine completely the spectrum of DQSs of index one and of PDQSs with the inside 4-cycle system of index one.     

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.     

Subquadrangle m‐regular systems on generalized quadrangles

We introduce the notion of subquadrangle regular system of a generalized quadrangle. A subquadrangle regular system of order m on a generalized quadrangle of order (s, t) is a set ? of embedded subquadrangles with the property that every point lies on exactly m subquadrangles of ?. If m is one half of the total number of subquadrangles on a point, we call ? a subquadrangle hemisystem. We construct two infinite families of symplectic subquadrangle hemisystems of the Hermitian surface ??(3, q²), q odd, and two infinite families of symplectic subquadrangle hemisystems of ??₃(q²), q even. Some sporadic examples of symplectic subquadrangle regular systems of ??(3, q²) are also presented. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:28‐41, 2010