设$\mathcal{D}$为非平凡$G$-块传递3-$(v,k,1)$设计，其中$T\leq G \leq \mathrm{Aut}(T)$，$T$为某有限非交换单群。证明若$T$是李型例外单群，则$T$必为Suzuki群${}^2B_2(q)$或$G_2(q)$。进一步，若$T={}^2B_2(q)$，则设计$\mathcal{D}$的参数为$v=q^2+1$与$k=q+1$，从而$\mathcal{D}$是阶为$q$的逆平面；若$T=G_2(q)$，则$T$中的点稳定子为$\mathrm{SL}_3(q).2$或$\mathrm{SU}_3(q).2$，且参数$k$满足严格受限条件。
英文摘要：
Let $\mathcal{D}$ be a non-trivial $G$-block-transitive $3$-$(v,k,1)$ design, where $T\leq G \leq \mathrm{Aut}(T)$ for some finite non-abelian simple group $T$. It is proved that if $T$ is a simple exceptional group of Lie type, then $T$ is either the Suzuki group ${}^2B_2(q)$ or $G_2(q)$. Furthermore, if $T={}^2B_2(q)$ then the design $\mathcal{D}$ has parameters $v=q^2+1$ and $k=q+1$, and so $\mathcal{D}$ is an inverse plane of order $q$; and if $T=G_2(q)$ then the point stabilizer in $T$ is either $\mathrm{SL}_3(q).2$ or $\mathrm{SU}_3(q).2$, and the parameter $k$ satisfies very restricted conditions.
In this article, we study $2$-designs with $\gcd(r,λ)=1$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type. We obtain four infinite families of such designs and provide some examples in each of these families.
This paper determined all pairs $(\mathcal{D},G)$ where $\mathcal{D}$ is a non-symmetric 2-$(v,k,λ)$ design with $(r,λ)=1$ and $G$ is the almost simple flag-transitive automorphism group of $\mathcal{D}$ with an exceptional socle of Lie type. We prove that if $T\trianglelefteq G\leq Aut(T)$ where $T$ is an exceptional group of Lie type, then $T$ must be the Ree group or Suzuki group, and there are five classes of non-isomorphic designs $\mathcal{D}$.
A code $C$ in the Hamming graph $\varGamma=H(m,q)$ is $2\it{\text{-neighbour-transitive}}$ if ${\rm Aut}(C)$ acts transitively on each of $C=C_0$, $C_1$ and $C_2$, the first three parts of the distance partition of $V\varGamma$ with respect to $C$. Previous classifications of families of $2$-neighbour-transitive codes leave only those with an affine action on the alphabet to be investigated. Here, $2$-neighbour-transitive codes with minimum distance at least $5$ and that contain "small" subcodes as blocks of imprimitivity are classified. When considering codes with minimum distance at least $5…
Let $q$ be a prime power and $V\cong{\mathbb F}_q^n$. A $t$-$(n,k,λ)_q$ 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 $V$, with the property that each $t$-dimensional subspace of $V$ is contained in precisely $λ$ elements of ${\mathcal B}$. Subspace designs are the $q$-analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group ${\rm Aut}({\mathcal D})$ acts transitively on ${\mathcal B}$. It is shown here that if $t\geq 2$ and…
Block-transitive Steiner $t$-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory, and cryptography. The main result of the paper settles an important open question: There exist no non-trivial examples with $t=7$ (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.
Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of ${\mathrm{PSL}}(2,q)$ and ${\mathrm{PΓL}}(2,q),$ whereas those of the third family are the affine solvable subgroups of ${\mathrm{AGL}}(2,q)$ found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exc…
Given integers $k$ and $m$, we construct a $G$-arc-transitive graph of valency $k$ and an $L$-arc-transitive oriented digraph of out-valency $k$ such that $G$ and $L$ both admit blocks of imprimitivity of size $m$.
A graph $\Ga$ is $G$-symmetric if $\Ga$ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\Ga$, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is imprimitive on $V(\Ga)$, namely when $V(\Ga)$ admits a nontrivial $G$-invariant partition $\BB$, the quotient graph $\Ga_{\BB}$ of $\Ga$ with respect to $\BB$ is always $G$-symmetric and sometimes even $(G, 2)$-arc transitive. (A $G$-symmetric graph is $(G, 2)$-arc transitive if $G$ is transitive on the set of oriented paths of length two.) In this paper we obtain nec…
This paper deals with block-transitive $t$-$(v,k,λ)$ designs in affine spaces for large $t$, with a focus on the important index $λ=1$ case. We prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-$(v,k,1)$ designs, except possibly when the group is one-dimensional affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.
In this article, we study $2$-designs with prime replication number admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine $2$-designs with prime replication number admitting an almost simple automorphism group.
We study the asymptotic behavior of a family of algebraic geometry codes, which we call block-transitive, that generalizes the classes of transitive and quasi-transitive codes. We prove, by using towers of algebraic function fields, that there are sequences of codes in this family attaining the Tsfasman-Vladut-Zink bound over finite fields of square cardinality. We give the exact length of these codes as well as explicit lower bounds for their parameters.
The purpose of this paper is to classify all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a non-trivial $2$-$(v, k, 2)$ design, and $G\leq Aut(\mathcal{D})$ acts transitively on the set of blocks of $\mathcal{D}$ and primitively on the set of points of $\mathcal{D}$ with sporadic socle. We prove that there exists only one such pair $(\mathcal{D}, G)$ in which $\mathcal{D}$ is a $2$-$(176,8,2)$ design and $G=HS$, the Higman-Sims simple group.
In this paper, we present a classification of $2$-designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,λ)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $AΓL_{1}(q)$.
We classify the pairwise transitive 2-designs, that is, 2-designs such that a group of automorphisms is transitive on the following five sets of ordered pairs: point-pairs, incident point-block pairs, non-incident point-block pairs, intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall into two classes: the symmetric ones and the quasisymmetric ones. The symmetric examples include the symmetric designs from projective geometry, the 11-point biplane, the Higman-Sims design, and designs of points and quadratic forms on symplectic spaces. The quasisymmetric examples aris…
We study point-block incidence structures $(\mathcal{P},\mathcal{B})$ for which the point set $\mathcal{P}$ is an $m\times n$ grid. Cameron and the fourth author showed that each block $B$ may be viewed as a subgraph of a complete bipartite graph $\mathbf{K}_{m,n}$ with bipartite parts (biparts) of sizes $m, n$. In the case where $\mathcal{B}$ consists of all the subgraphs isomorphic to $B$, under automorphisms of $\mathbf{K}_{m,n}$ fixing the two biparts, they obtained necessary and sufficient conditions for $(\mathcal{P},\mathcal{B})$ to be a $2$-design, and to be a $3$-design. We first re-i…
In this article, we investigate $2$-$(v,k,λ)$ designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,λ)$ design with $\gcd(k,λ)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq AΓL_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the …
Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters, m and n, now called Delandtsheer--Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsheer--Doyen parameters provide upper bounds on the permutation ranks of the groups induced on the imprimitivity system and on a class of the system. We explore extreme cases where these bounds are attained, give a new construction for a family of d…
In this article, we study $2$-designs with $\gcd(r, λ)=1$ admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a $2$-design with $\gcd(r, λ)=1$ and $G$ is a flag-transitive almost simple automorphism group of $\mathcal{D}$ whose socle is $X=\mathrm{PSU}(n, q)$ with $(n, q)\neq (3, 2)$ and prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt-Bose-Shrikhande spaces, or it is isomorphic to a design …
A graph $Γ$ 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 $Γ$ with $V(Γ)$ admitting a nontrivial $G$-invariant partition $\mathcal{B}$ such that there is exactly one edge of $Γ$ between any two distinct blocks of $\mathcal{B}$. This is achieved by giving a classification of $(G, 2)$-point-transitive and $G$-block-transitive designs $\mathcal{D}$ together with $G$-orbits $Ω$ on the flag set of $\mathcal{D}$ such that $G_{σ, L}$ is transitive on $L \setmin…
It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group $G$ of a $2$-$(k^{2}, k, λ)$ design D, with $λ\mid k$, is either an affine group or an almost simple classical group. Moreover, when $G$ is the smallest Ree group, $\mathcal{D}$ is isomorphic either to the $2$-$(62, 6, 2)$ design or to one of the three $2$- $(62, 6, 6)$ designs constructed in this paper. All the four $2$-designs have the $36$ secants of a nondegenerate conic $\mathcal{C}$ of $PG_{2}(8)$ as a point set and 6-sets of secants in a remarkable configuration as a block set.
This paper initiates the investigation of the family of $(G,s)$-geodesic-transitive digraphs with $s\geq 2$. We first give a global analysis by providing a reduction result. Let $Γ$ be such a digraph and let $N$ be a normal subgroup of $G$ maximal with respect to having at least $3$ orbits. Then the quotient digraph $Γ_N$ is $(G/N,s')$-geodesic-transitive where $s'=\min\{s,\diam(Γ_N)\}$, $G/N$ is either quasiprimitive or bi-quasiprimitive on $V(Γ_N)$, and $Γ_N$ is either directed or an undirected complete graph. Moreover, it is further shown that if $Γ$ is not $(G,2)$-arc-transitive, then $G/N…
In 1987, Huw Davies proved that, for a flag-transitive point-imprimitive $2$-$(v,k,λ)$ design, both the block-size $k$ and the number $v$ of points are bounded by functions of $λ$, but he did not make these bounds explicit. In this paper we derive explicit polynomial functions of $λ$ bounding $k$ and $v$. For $λ\leq 4$ we obtain a list of `numerically feasible' parameter sets $v, k, λ$ together with the number of parts and part-size of an invariant point-partition and the size of a nontrivial block-part intersection. Moreover from these parameter sets we determine all examples with fewer than …
Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where $G_v$ is a stabilizer of $v\in V$ and ${\cal F}$ runs over all intersecting sets in $G$. Intersection densities of transitive groups of degree $pq$, where $p>q$ are odd primes, is considered. In particular, t…
Let $G=(V,E)$ be a simple connected graph. A set of vertices $S\subseteq V$ is said to be a dominating set if for any vertex in $V\setminus S$ is adjacent to at least one vertex in $S$. The domination number $γ(G)$ of $G$ is the minimum cardinality among all such sets. In this paper, we obtain some results on the domination number of the incidence graphs of combinatorial designs. In particular, we prove a conjecture and disprove another conjecture in a recent paper by Goldberg, Rajendraprasad and Mathew. We also prove a third conjecture by the same authors for block-transitive symmetric design…
We study the first uniformly finite homology group of Block and Weinberger for uniformly locally finite graphs, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_2$. When the graph is a tree, or coefficients are in $\mathbb{Z}_2$, a characterisation of the group is obtained. In the general case, we describe three phenomena that entail non-vanishing of the group; their disjunction is shown to also be necessary for non-vanishing in the case of transitive graphs.
It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme …
In this expository paper we describe a group theoretic characterization of arc-transitive graphs and their quotients. When passing from a symmetric graph to its quotient, much information is lost, but some of this information may be recovered from a certain combinatorial design on the blocks, as well as a bipartite graph between the blocks. We address the "extention problem" which asks, when is this additional information sufficient to recover the original graph?
For a finite group $G$ and an inverse-closed generating set $C$ of $G$, let $Aut(G;C)$ consist of those automorphisms of $G$ which leave $C$ invariant. We define an $Aut(G;C)$-invariant normal subgroup $Φ(G;C)$ of $G$ which has the property that, for any $Aut(G;C)$-invariant normal set of generators for $G$, if we remove from it all the elements of $Φ(G;C)$, then the remaining set is still an $Aut(G;C)$-invariant normal generating set for $G$. The subgroup $Φ(G;C)$ contains the Frattini subgroup $Φ(G)$ but the inclusion may be proper. The Cayley graph $Cay(G,C)$ is normal edge-transitive if $A…
This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that ${\rm{Aut}}(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes \textsf{D}_{n+1},$ where $\textsf{N}$ is the subgroup fixing $S_n$ element-wise and $\textsf{D}_{n+1}$ is a dihedral group of order $2(n+1)$. We conjecture that $\textsf{N}$ is trivial. We also prove that the subgraph $Γ$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose aut…
A directed triple system of order $v$ (or, DTS$(v)$) is decomposition of the complete directed graph $\vec{K_v}$ into transitive triples. A $v$-good sequencing of a DTS$(v)$ is a permutation of the points of the design, say $[x_1 \; \cdots \; x_v]$, such that, for every triple $(x,y,z)$ in the design, it is not the case that $x = x_i$, $y = x_j$ and $z = x_k$ with $i < j < k$. We prove that there exists a DTS$(v)$ having a $v$-good sequencing for all positive integers $v \equiv 0,1 \bmod {3}$. Further, for all positive integers $v \equiv 0,1 \bmod {3}$, $v \geq 7$, we prove that there is a DTS…
A graph $Γ$ is $G$-symmetric if $G$ is a group of automorphisms of $Γ$ which is transitive on the set of ordered pairs of adjacent vertices of $Γ$. If $V(Γ)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks $B, C \in {\cal B}$ adjacent in the quotient graph $Γ_{\cal B}$ of $Γ$ relative to ${\cal B}$, exactly one vertex of $B$ has no neighbour in $C$, then $Γ$ is called an almost multicover of $Γ_{\cal B}$. In this case an incidence structure with point set ${\cal B}$ arises naturally, and it is a $(G, 2)$-point-transitive and $G$-block-transitive 2-design if in addit…
Segerman's 15+4 puzzle is a hinged version of the classic 15-puzzle, in which the tiles rotate as they slide around. In 1974, Wilson classified the groups of solutions to sliding block puzzles. We generalize Wilson's result to puzzles like the 15+4 puzzle, where the tiles can rotate, and the sets of solutions are subgroups of the generalized symmetric groups. Aside from two exceptional cases, we see that the group of solutions to such a puzzle is always either the entire generalized symmetric group or one of two special subgroups of index two.
It is possible to define mixing properties for subshifts according to the intensity which allows to concatenate two rectangular blocks. We study the interplay between this intensity and computational properties. In particular we prove that there exists linearly block gluing subshift of finite type which are aperiodic and that all right-recursively enumerable positive number can be realized as entropy of linearly block gluing Z 2-subshift of finite type. Like linearly block gluing imply transitivity, this last point answer a question asked in [HM10] about the characterization of the entropy of …
