A note on relative hemisystems of Hermitian generalised quadrangles

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

We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.

Generalised quadrangles and transitive pseudo-hyperovals

A pseudo-hyperoval of a projective space $\PG(3n-1,q)$, $q$ even, is a set of $q^n+2$ subspaces of dimension $n-1$ such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is elementary. We then deduce from this result a classification of the thick generalised quadrangles $\mathcal{Q}$ that admit a point-primitive, line-transitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that $\mathcal{Q}$ is flag-transitive and isomorphic to $T_2^*(\mathcal{H})$, where $\mathcal{H}$ is either the regular …

Neighbour-Transitive Codes and Partial Spreads in Generalised Quadrangles

A code $C$ in a generalised quadrangle ${\mathcal Q}$ is defined to be a subset of the vertex set of the point-line incidence graph $\varGamma$ of ${\mathcal Q}$. The minimum distance $δ$ of $C$ is the smallest distance between a pair of distinct elements of $C$. The graph metric gives rise to the distance partition $\{C,C_1,\ldots,C_ρ\}$, where $ρ$ is the maximum distance between any vertex of $\varGamma$ and its nearest element of $C$. Since the diameter of $\varGamma$ is $4$, both $ρ$ and $δ$ are at most $4$. If $δ=4$ then $C$ is a partial ovoid or partial spread of ${\mathcal Q}$, and if, …

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).

No almost simple sporadic group acts primitively on the points of a generalised quadrangle

A generalised quadrangle is a point-line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits, and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as au…

Point-primitive, line-transitive generalised quadrangles of holomorph type

Let $G$ be a group of collineations of a finite thick generalised quadrangle $Γ$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of $Γ$, and transitively on the lines of $Γ$. We show that the primitive action of $G$ on $\mathcal{P}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples $Γ$ for which the action of $G$ on $\mathcal{P}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where ther…

Simple groups, product actions, and generalised quadrangles

The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of…

On codewords in the dual code of classical generalised quadrangles and classical polar spaces

In [9], the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q+(5, q) and H(5, q2), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and char…

厄米特曲面\\(\mathsf{H}(3,q^2)\\)的每个特殊集都是经典的

Every special set of the Hermitian surface $\mathsf{H}(3,q^2)$ is classical

研究厄米特曲面\\(\mathsf{H}(3,q^2)\\)特殊集，通过证明其特殊集仅已知的一个示例，该示例产生经典广义四边形，从而得出不存在其他特殊集的结论。