Related papers: Flag-transitive $2$-$(v,k,\lambda)$ designs with $…

A $3$-$(v,\{4,6\},1)$ design is a configuration of $v$ points and a collection of $4$- and $6$-element subsets called blocks, that jointly contain every 3-element subset exactly once. Using an exhaustive computer search on $v\leq 28$ points…

A connected graph $\Gamma=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of…

We announce the classification of two related classes of flag-transitive geometries. There is an infinite family of such geometries, related to the nonsplit extensions $3^{[{n\atop 2}]_{_2}}\cdot \SP_{2n}(2)$, and twelve sporadic examples…

Let $q$ be a prime power and $V\cong{\mathbb F}_q^n$. A $t$-$(n,k,\lambda)_q$ design, or simply a subspace design, is a pair ${\mathcal D}=(V,{\mathcal B})$, where ${\mathcal B}$ is a subset of the set of all $k$-dimensional subspaces of…

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper classifies those lines that exhibit almost simple symmetries. We introduce a general recipe…

In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using…

We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding…

We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple…

Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with six families of groups which are $2$-transitive, or Frobenius, or one-dimensional affine, or the affine solvable subgroups of $ \mathrm{AGL}(2, q)$, or…

A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the…

We study $G$-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups $G$ with socle $\mathrm{PSL}_n(q)$. It turns out that $s\leqslant2$ for such digraphs, which provides the first step in determining an upper bound on…

Recently, Leemans and Stokes constructed an infinite family of incidence geometries admitting trialities but no dualities from the groups PSL(2,q) (where $q=p^{3n}$ with $p$ a prime and $n>0$ a positive integer). Unfortunately these…

A locally primitive 2-design is a 2-design admitting an automorphism group $G$ with primitive local actions. It is proved that $G$ is point-primitive, and either $G$ is an almost simple group, or $G$ acting on the points is an affine group.

Let A be an evolution algebra (possibly infinite-dimensional) equipped with a fixed natural basis B, and let E be the associated graph defined by Elduque and Labra. We describe the group of automorphisms of A that are diagonalizable with…

Let $N=L_n(q)$, {$n \geq 2$}, $q$ a prime power, be a projective linear simple group. We classify all Steiner quadruple systems admitting a group $G$ with $N \leq G \leq \Aut(N)$. In particular, we show that $G$ cannot act as a group of…

Group action is a standard approach to obtain $t$-designs. In this approach, selecting a specific permutation group with a certain degree of transitivity or homogeneity and a proper set of base blocks is important for obtaining $t$-$(v, k,…

We study vertex-quasiprimitive $2$-arc-transitive digraphs, reducing the problem of vertex-primitive $2$-arc-transitive digraphs to almost simple groups. This includes a complete classification of vertex-quasiprimitive $2$-arc-transitive…

A non-complete graph $\Gamma$ is said to be $(G,2)$-distance transitive if $G$ is a subgroup of the automorphism group of $\Gamma$ that is transitive on the vertex set of $\Gamma$, and for any vertex $u$ of $\Gamma$, the stabilizer $G_u$ is…

We extend the notion of quasi-transitive orientations of graphs to 2-edge-coloured graphs. By relating quasi-transitive $2$-edge-colourings to an equivalence relation on the edge set of a graph, we classify those graphs that admit a…

A graph $\Gamma$ is called $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of $G$-symmetric graphs $\Gamma$ with $V(\Gamma)$ admitting…