Generalized Quadrangles with a Spread of Symmetry

We present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint.     

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.     

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.     

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).     

Translation Generalized Quadrangles In Even Characteristic

In this paper, we first introduce new objects called “translation generalized ovals” and “translation generalized ovoids”, and make a thorough study of these objects. We then obtain numerous new characterizations of the of Tits and the classical generalized quadrangle in even characteristic, including the complete classification of 2-transitive generalized ovals for the even case. Next, we prove a new strong characterization theorem for the of Tits. As a corollary, we obtain a purely geometric proof of a theorem of Johnson on semifield flocks. * The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).     

Ovoids and Bipartite Subgraphs in Generalized Quadrangles

A point-line incidence system is called an -partial geometry of order (s,t) if each line contains s + 1 points, each point lies on t + 1 lines, and for any point a not lying on a line L, there exist precisely lines passing through a and intersecting L (the notation is pG (s,t)). If = 1, then such a geometry is called a generalized quadrangle and denoted by GQ(s,t). It is established that if a pseudogeometric graph for a generalized quadrangle GQ(s,s ² – s) contains more than two ovoids, then s = 2. It is proved that the point graph of a generalized quadrangle GQ(4,t) contains no K _4,6-subgraphs. Finally, it is shown that if some -subgraph of a pseudogeometric graph for a generalized quadrangle GQ(4,t) contains a triangle, then t 6.     

Groups of Projectivities of Generalized Quadrangles

We characterize some classical quadrangles by means of properties of their groups of projectivities. In particular, we characterize all finite classical quadrangles with regular lines, and all symplectic quadrangles over quadratically closed fields.     

Translation Ovoids of Generalized Quadrangles and Hexagnos

We define the notion of a translation ovoid in the classical generalized quadrangles and hexagons of order q, and we enumerate all known examples; translation spreads are defined dually. A modification of the known ovoids in the generalized hexagon H(q), q=3^2h+1, yields new ovoids of that hexagon. Dualizing and projecting along reguli, we obtain an alternative construction of the Roman ovoids due to Thas and Payne. Also, we construct a new translation spread in H(q) for any 1 mod 3, q odd, with the property that any projection along reguli yields the classical ovoid in the generalized quadrangle Q(4,q). Finally, we prove that for q odd, the new example is the only non-Hermitian translation spread in H(q) with the property that any projection along reguli yields the classical ovoid in Q(4,q).     

Pseudodual Grids and Extensions of Generalized Quadrangles

Siberian Mathematical Journal -     

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.     

Distance Transitive Generalized Quadrangles of Prime Order

Without using the classification of finite simple groups, we classify the finite generalized quadrangles of prime order admitting a group acting distance transitively on the collinearity graph. Our method uses combinatorial geometry and permutation groups.     

Generalized Quadrangles with an Abelian Singer Group

In this note we characterize thick finite generalized quadrangles constructed from a generalized hyperoval as those admitting an abelian Singer group, i.e., an abelian group acting regularly on the points. S. De Winter: The first author is a Research Assistant of the Fund for Scientific Research—Flanders (Belgium). K. Thas: The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).     

On Near Hexagons and Spreads of Generalized Quadrangles

The glueing-construction described in this paper makes use of two generalized quadrangles with a spread in each of them and yields a partial linear space with special properties. We study the conditions under which glueing will give a near hexagon. These near hexagons satisfy the nice property that every two points at distance 2 are contained in a quad. We characterize the class of the glued near hexagons and give examples, some of which are new near hexagons.     

Local Sharply Transitive Actions on Finite Generalized Quadrangles

We classify the finite generalized quadrangles containing a line L such that some group of collineations acts sharply transitively on the ordered pentagons which start with two points of L. This can be seen as a generalization of a result of Thas and the second author [22] classifying all finite generalized quadrangles admitting a collineation group that acts transitively on all ordered pentagons, although the restriction to sharp transitivity is essential in our arguments. However, the conclusion is exactly the same family of classical generalized quadrangles (the orthogonal quadrangles and their duals). Our main result thus provides a local group theoretic characterization of these classical quadrangles.     

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

A pseudo‐hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo‐hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that is flag‐transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).     

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.     

广义对称性在积分计算中的应用

本文将积分计算中的对称性方法推广到了一般情形 ,并提出了通过适当改造被积函数以利用对称性来简化计算的方法 .