Generalized Hexagons as Amalgamations of Generalized Quadrangles

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 n...

Coordinatization structures for generalized quadrangles and glued near hexagons

A generalized admissible triple is a triple T = (L, X,∆), where X is a set of size s + 1 ≥ 2, L is a Steiner system S(2, s + 1, st + 1), t ≥ 1, with point-set P and ∆ is a very nice map from P × P to the group Sym(X) of all permutations of the set X. Generalized admissible triples can be used to coordinatize generalized quadrangles with a regular spread and glued near hexagons. The idea of coor...

Characterizations of Generalized Quadrangles by Generalized Homologies

Let S= (P, B, I) be a generalized quadrangle of order (s, t). For X, y E P, X+ y. let X(x, y) be the group of all collineations of S fixing x and y linewise. If ZE (4 YJL> then the set of all points incident with the line xz (resp. y;) is denoted by z(resp. 3. The generalized quadrangle S = (P, B, I) is said to be (x, y)-transitive, x+ y, if X(x, y) is transitive on each set Z{x, z} and jZ’-{ y...

Finite Generalized Quadrangles

Foundations of elation generalized quadrangles

In these notes we are interested in the following fundamental question: “Given a thick generalized quadrangle S(x) with elation point (respectively center of transitivity) x , when does the set of all elations about x form a group?”. It was the general belief for a long time that this is usually the case. However, the goal of these notes is to show that this belief is wrong. To that end, we wil...