Every generalized quadrangle of order 5 having a regular point is symplectic

For many years now, one of the most important open problems in the theory of generalized quadrangles has been whether other classes of generalized quadrangles exist besides those that are currently known. This paper reports on an unsuccessful attempt to construct a new generalized quadrangle. As a byproduct of our attempt, however, we obtain the following new characterization result: every generalized quadrangle of order 5 that has at least one regular point is isomorphic to the quadrangle W(5) arising from a symplectic polarity of PG(3, 5). During the classification process, we used the computer algebra system GAP to perform certain computations or to search for an optimal strategy for the proof

A construction of small (q-1)-regular graphs of girth 8

In this note we construct a new infinite family of $(q-1)$-regular graphs of girth $8$ and order $2q(q-1)^2$ for all prime powers $q\ge 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.Comment: 8 pages, 2 figure

A formulation of a (q+1,8)-cage

Let $q\ge 2$ be a prime power. In this note we present a formulation for obtaining the known $(q+1,8)$-cages which has allowed us to construct small $(k,g)$--graphs for $k=q-1, q$ and $g=7,8$. Furthermore, we also obtain smaller $(q,8)$-graphs for even prime power $q$.Comment: 14 pages, 2 figure

Exceptional Moufang quadrangles and structurable algebras

In 2000, J. Tits and R. Weiss classified all Moufang spherical buildings of rank two, also known as Moufang polygons. The hardest case in the classification consists of the Moufang quadrangles. They fall into different families, each of which can be described by an appropriate algebraic structure. For the exceptional quadrangles, this description is intricate and involves many different maps that are defined ad hoc and lack a proper explanation. In this paper, we relate these algebraic structures to two other classes of algebraic structures that had already been studied before, namely to Freudenthal triple systems and to structurable algebras. We show that these structures give new insight in the understanding of the corresponding Moufang quadrangles.Comment: 49 page

Ree geometries

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines

A new construction of Moufang quadrangles of type E6, E7 and E8

In the classification of Moufang polygons by J. Tits and R. Weiss, the most intricate case is by far the case of the exceptional Moufang quadrangles of type E6, E7 and E8, and in fact, the construction that they present is ad-hoc and lacking a deeper explanation. We will show how tensor products of two composition algebras can be used to construct these Moufang quadrangles in characteristic different from 2. As a byproduct, we will obtain a method to construct any Moufang quadrangle in characteristic different from two from a module for a Jordan algebra

On generalized quadrangles with some concurrent axes of symmetry

Let S be a finite Generalized Quadrangle (GQ) of order (s, t), s not equal 1 not equal t, and suppose L is a line of S. A symmetry about L is an automorphism of S which fixes every line concurrent with L. A line L is an axis of symmetry if there is a full group of size s of symmetries about L. A point of a generalized quadrangle is a translation point if every line through it is an axis of symmetry. If there is a point p in a GQ S = (P, B, 1) for which there is a group G of automorphisms of the GQ which fixes p linewise, and such that G acts regularly on the points of P \ p(perpendicular to), then S is called an elation generalized quadrangle, and instead of S, often the notations (S-(P), G) or S(P) are used. If G is abelian, then (S-(P), G) is a translation generalized quadrangle (TGQ), and a GQ is a TGQ S-(P) if and only if p is a translation point, see [9]. We study the following two problems. (1) Suppose S is a G Q of order (8, t) s not equal 1 not equal t. How many distinct axes of symmetry through the same point p are needed to conclude that every line through p is an axis of symmetry, and hence that S-(P) is a TGQ? (2) Given a TGQ (S-(P), G), what is the minimum number of distinct lines through p such that G is generated by the symmetries about these lines