SIMPLE GROUPS, PRODUCT ACTIONS, AND GENERALIZED QUADRANGLES

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 pentago...

Certain Roman and flock generalized quadrangles have nonisomorphic elation groups

We prove that the elation groups of a certain infinite family of Roman generalized quadrangles are not isomorphic to those of associated flock generalized quadrangles.

Finite Generalized Quadrangles

Monomial graphs and generalized quadrangles

Let Fq be a finite field, where q = p for some odd prime p and integer e ≥ 1. Let f, g ∈ Fq[x, y] be monomials. The monomial graph Gq(f, g) is a bipartite graph with vertex partition P ∪L, P = Fq = L, and (x1, x2, x3) ∈ P is adjacent to [y1, y2, y3] ∈ L if and only if x2 + y2 = f(x1, y1) and x3 + y3 = g(x1, y1). Dmytrenko, Lazebnik, and Williford proved in [6] that if p ≥ 5 and e = 23 for integ...

Flocks, ovoids and generalized quadrangles

Let (∞, π∞) be an incident point-plane pair of PG(3, q). A tetrad with respect to (∞, π∞) is a set {X,Y, Z,W} of points of PG(3, q) \ π∞ such that {∞, X, Y, Z,W} is a cap of PG(3, q), ∞ ∈ 〈X,Y, Z〉 and W 6∈ 〈X,Y, Z〉. A set Θ of ovoids of PG(3, q) is tetradic with respect to (∞, π∞) if each ovoid contains ∞, has tangent plane π∞ and is such that each tetrad with respect to (∞, π∞) is contained in...