Generalised quadrangles with a group of automorphisms acting primitively on points and lines

Compact Generalised Quadrangles with Parameter 1 and Large Group of Automorphisms

Intriguing Sets of Partial Quadrangles

Finite geometry is the study of incidence structures, a system of points and lines with some points on some lines according to an incidence relation, such that the number of points and lines is finite. Generalised quadrangles arose from an effort by Tits [5] to give geometric constructions of the finite simple groups of Lie type. Tits' key observation was that the incidence graph of an n-gon al...

The Classification of Finite Generalized Quadrangles Admitting a Group Acting Transitively on Ordered Pentagons

Let S be a thick generalized quadrangle and let G be a group of automorphisms of S. If G acts transitively on the set of non-degenerate ordered pentagons, then S is one of the classical generalized quadrangles W (q), Q(4, q), Q(5, q) or H(3, q2). The possibilities for G in each case are determined. We do not use the classification of the finite simple groups (from which this result also follows).

Intriguing sets of partial quadrangles

The point-line geometry known as a partial quadrangle (introduced by Cameron in 1975) has the property that for every point/line non-incident pair (P, `), there is at most one line through P concurrent with `. So in particular, the well-studied objects known as generalised quadrangles are each partial quadrangles. An intriguing set of a generalised quadrangle is a set of points which induces an...

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.