Generalized quadrangles associated with G2(q)

Generalized Quadrangles Associated with G2(q)

In addition to the generalized quadrangles arising naturally from smalldimensional symplectic, unitary, or orthogonal geometries, the only known finite examples are the following quadrangles and their duals: ones with parameters 4, 4 (where 4 = 2e > 8) or q*, 4 (where q = 22e+l > 8) due to Tits [4, p. 3041; and others with parameters q 1, q + 1 (for prime powers q) due to Hall [5] and Ahrens an...

Finite Generalized Quadrangles

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...

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...

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.