A Kantor family is a collection of subgroups from which a generalized quadrangle can be constructed using Kantor ' s idea. This paper considers the case in which some of the subgroups in the Kantor family or its related family are normal in the ambient finite group G. We show that if two members of a Kantor family are normal in G then G is elementary abelian and that if all members of the relat...

An octagon quadrangle is the graph consisting of an 8-cycle (x1, x2, ..., x8) with two additional chords: the edges {x1, x4} and {x5, x8}. An octagon quadrangle system of order v and index λ [OQS] is a pair (X,H), where X is a finite set of v vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of λKv defined on X. An octagon quadrangl...

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...

An intriguing set of points of a generalised quadrangle was introduced in [2] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [2] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi point...

We determine the p-rank of the point-line incidence matrix of the generalized quadrangle of type Sp(4, p) where p is prime.

A truly fruitful way to construct finite generalized quadrangles is through the detection of Kantor families in the general 5-dimensional Heisenberg group H2(q) over some finite field Fq . All these examples are so-called “flock quadrangles”. Payne (Geom. Dedic. 32:93–118, 1989) constructed from the Ganley flock quadrangles the new Roman quadrangles, which appeared not to arise from flocks, but...