On Eggs and Translation Generalised Quadrangles

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...

Weighted intriguing sets of finite generalised quadrangles

We construct and analyse interesting integer valued functions on the points of a generalised quadrangle which lie in the orthogonal complement of a principal eigenspace of the collinearity relation. These functions generalise the intriguing sets introduced by Bamberg et al. (Combinatorica 29(1):1–17, 2009), and they provide the extra machinery to give new proofs of old results and to establish ...

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...

Compact Generalised Quadrangles with Parameter 1 and Large Group of Automorphisms

Elation generalised quadrangles of order (s, p), where p is prime

We show that an elation generalised quadrangle which has p + 1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle).