A finite linear space is a finite set of points and lines, where any two points lie on a unique line. Well known examples include projective planes. This project focuses on linear spaces which admit certain types of symmetries. Symmetries of the space which preserve the line structure are called automorphisms. A group of these is called an automorphism group of the linear space. Two interesting properties of linear spaces are point imprimitivity and line transitivity. Point imprimitive spaces admit a second structure on the points aside from the lines, which is also preserved by an automorphism group. In line transitive spaces, given any two lines, an automorphism can be found that maps one line to the other.
Very few point imprimitive, line transitive linear spaces, apart from projective planes, are known. Such spaces that have been found have been surprising. One point of interest is whether such spaces are rare and the known ones are in some sense exceptional, or if there are many such spaces, but mathematicians have been looking in the wrong places.
Here we investigate methods to construct a line transitive, point imprimitive linear space over a given point set and automorphism group. We employ these methods on two given automorphism groups, both on a set of 451 points. This was an exceptional situation identified in theoretical work of Praeger and Tuan. Included in this is the development of an algorithm, written in GAP, an algebraic programming system, and C, to perform these constructions. This algorithm is extendible to a wider class of groups.
The symmetric $2$-$(v,k,λ)$ designs, with $k>λ\left(λ-3 \right)/2$, admitting a flag-transitive, point-imprimitive automorphism group are completely classified: they are the known $2$-designs with parameters $(16,6,2),(45,12,3),(15,8,4)$ or $(96,20,4)$.
In this paper we complete a classification of finite linear spaces $\cS$ with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders $4,7, 9$ and 11, two designs on 91 points with line size 6, and 467 designs on 729 points with line size 8.
Let $\mathcal{D}=\left(\mathcal{P},\mathcal{B} \right)$ be a symmetric $2$-$(v,k,λ)$ design admitting a flag-transitive, point-imprimitive automorphism group $G$ that leaves invariant a non-trivial partition $Σ$ of $\mathcal{P}$. Praeger and Zhou \cite{PZ} have shown that, there is a constant $k_{0}$ such that, for each $B \in \mathcal{B}$ and $Δ\in Σ$, the size of $\left\vert B \cap Δ\right \vert$ is either $0$ or $k_{0}$. In the present paper we show that, if $k>λ\left(λ-3 \right)/2$ and $k_{0} \geq 3$, $\mathcal{D}$ is isomorphic to one of the known flag-transitive, point-imprimitive symmet…
In this paper we develop several general methods for analysing flag-transitive point-imprimitive $2$-designs, which give restrictions on both the automorphisms and parameters of such designs. These constitute a tool-kit for analysing these designs and their groups. We apply these methods to complete the classification of flag-transitive, point-imprimitive $2$-$(v,k,λ)$ designs with $λ$ at most $4$.
This paper is a further contribution to the classification of line-transitive finite linear spaces. We prove that if S is a non-trivial finite linear space with the Fang-Li parameter gcd(k,r) is 9 or 10, the automorphism group G of S is line-transitive and point-imprimitive, then S is the Desarguesian projective plane PG(2,9).
We give a construction of a family of designs with a specified point-partition, and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric $2$-$(1408,336,80)$ design with automorphism group $2^{12}:((3\cdot\mathrm{M}_{22}):2)$, and a construction of one of the families of the symplectic designs (the designs $S^-(n)$) exhibit…
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 …
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…
A retract of a graph $Γ$ is an induced subgraph $Ψ$ of $Γ$ such that there exists a homomorphism from $Γ$ to $Ψ$ whose restriction to $Ψ$ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph $Γ$ is $G$-symmetric if $G$ is a subgroup of the automorphism group of $Γ$ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of $Γ$ admits a nontrivial partition that …
In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point-imprimitive group action, for a given value of gcd(k,r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k,r) at most 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of…
This paper gives a sufficient and necessary condition for the existence of an (X, s)-arc-transitive imprimitive graph which is not a multicover of a given quotient graph.
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…
We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph $D(Γ)$ of a given graph $Γ$ to show that the uniform vertex-transitivity of $Γ$ is equivalent to the existence of cliques of sufficient size in $D(Γ)$. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient criteria for uniform vertex-tra…
Unitary graphs are arc-transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent classification of a class of arc-transitive graphs that admit an automorphism group acting imprimitively on the vertices. In this paper we prove that all unitary graphs are connected of diameter two and girth three. Based on this we obtain, for any prime power $q > 2$, a lower bound of order $O(Δ^{5/3})$ on the maximum number of vertices in an arc-transitive graph of degree $Δ= q(q^2-1)$ and diame…
This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than delta. We show that with possibly finitely many exceptions, one can take delta = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements goes to 1 as n increases and prove similar results for families of permutation representati…
Let $G$ be a flag-transitive automorphism group of a $(v,k,λ)$ symmetric design $\mathcal{D}$ with $k>λ(λ-2)$. O'Reilly Regueiro proved that if $G$ is point-imprimitive, then $\mathcal{D}$ has parameters $(v,k,λ)=(λ^2(λ+2),λ(λ+1),λ)$. In the present paper, we consider the case that $G$ is point-primitive. By applying the O'Nan-Scott Theorem, we prove that $G$ must be of affine type or almost simple type.
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…
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…
For n at least 2, the concept of n-way expanders was defined by various researchers. Bigger n gives a weaker notion in general, and 2-way expanders coincide with expanders in usual sense. Koji Fujiwara asked whether these concepts are equivalent to that of ordinary expanders for all n for a sequence of Cayley graphs. In this paper, we answer his question in the affirmative. Furthermore, we obtain universal inequalities on multi-way isoperimetric constants on any finite connected vertex-transitive graph, and show that gaps between these constants imply the imprimitivity of the group action on t…
The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset Γsuch that the subgroup G of graph automorphisms leaving Γinvariant is transitive on both the set Γof `codewords' and also the set of `neighbours' of Γ, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group G is a subgroup of the symmetric group on V and is intransitive or imprimitive on the underlying v-set V. In the r…
A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(Ω)$ is intersecting if for any $g,h\in \mathcal{F}$ there exists $ω\in Ω$ such that $ω^g = ω^h$. The \emph{intersection density} $ρ(G)$ of $G$ is the maximum of $\left\{ \frac{|\mathcal{F}|}{|G_ω|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}$, where $G_ω$ is the stabilizer of $ω$ in $G$. In this paper, it is proved that if $G$ is an imprimitive group of degree $pq$, where $p$ and $q$ are distinct odd primes, with at least two systems of imprimitivity then $ρ(G) = 1$. Moreover, if $G$ is primitive of de…
Let $Γ=(V,E)$ be a graph. The square graph $Γ^2$ of the graph $Γ$ is the graph with the vertex set $V(Γ^2)=V$ in which two vertices are adjacent if and only if their distance in $Γ$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, w…
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 basic geometry was shown to involve a faithful group action, which is primitive or quasiprimitive respectively, on the set of elements of each type. For each O'Nan-Scott type of primitive group, we construct a new infinite family of geometries, which are thick and of unbounded rank, and which admit a flag-transitive automorphism group acting faithfully on the s…
Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a key role in a recent approach to exploit symmetry in integer convex optimization problems. Here, naturally the question arises, for which groups the number of core points is finite up to translations by vectors fixed by the group. In this paper we consider transitive permutation groups and prove this type of finiteness for the $2$-homogeneous ones. We provid…
