Related papers: Simple groups, product actions, and generalised qu…

Let $G$ be a group of collineations of a finite thick generalised quadrangle $\Gamma$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of $\Gamma$, and transitively on the lines of $\Gamma$. We show that the primitive…

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.

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…

In 2008, Schneider and Van Maldeghem proved that if a group acts flag-transitively, point-primitively, and line-primitively on a generalised hexagon or generalised octagon, then it is an almost simple group of Lie type. We show that…

Suppose that an automorphism group $G$ acts flag-transitively on a finite generalized hexagon or octagon $\cS$, and suppose that the action on both the point and line set is primitive. We show that $G$ is an almost simple group of Lie type,…

Let $G$ be a collineation group of a thick finite generalised hexagon or generalised octagon $\Gamma$. If $G$ acts primitively on the points of $\Gamma$, then a recent result of Bamberg et al. shows that $G$ must be an almost simple group…

In earlier work we gave a characterisation of pregeometries which are `basic' (that is, admit no `non-degenerate' quotients) relative to two different kinds of quotient operations, namely imprimitive quotients and normal quotients. Each…

We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie…

Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $\pi$ acts $2$-transitively on the points of $\pi$, then $\pi$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to…

Let $\mathcal{D}$ be a non-trivial quasi-symmetric $2$-design with two block intersection numbers $x=0$ and $2\leq y\leq10$, and suppose that $G$ is an automorphism group of $\mathcal{D}$. If $G$ is flag-transitive and point-primitive, then…

Let $G$ be a flag-transitive automorphism group of a $(v,k,\lambda)$ symmetric design $\mathcal{D}$ with $k>\lambda(\lambda-2)$. O'Reilly Regueiro proved that if $G$ is point-imprimitive, then $\mathcal{D}$ has parameters…

Various descending chains of subgroups of a finite permutation group can be used to define a sequence of `basic' permutation groups that are analogues of composition factors for abstract finite groups. Primitive groups have been the…

In this article, we investigate symmetric designs admitting a flag-transitive and point-primitive affine automorphism group. We prove that if an automorphism group $G$ of a symmetric $(v,k,\lambda)$ design with $\lambda$ prime is…

Let $G$ be a semisimple algebraic group whose decomposition into a product of simple components does not contain simple groups of type $A$, and $P\subseteq G$ be a parabolic subgroup. Extending the results of Popov [7], we enumerate all…

A generalized quadrangle is a point-line incidence geometry $\mathcal{Q}$ such that: (i) any two points lie on at most one line, and (ii) given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$…

According to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a…

A central problem in the study of generalized quadrangles is to classify finite generalized quadrangles satisfying certain symmetry conditions. It is known that an automorphism group of a finite thick generalized quadrangle $\mathcal{S}$…

Let $\mathcal{D} = (\mathcal{P}, \mathcal{B})$ be a $2$-$(v, k, \lambda)$ design, and let $G$ be a half-flag-transitive automorphism group of ${\cal D}$. In this article, we first establish three sufficient conditions for $G$ to be…

A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a…

In this article, we study symmetric designs admitting flag-transitive, point-imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting…