arXiv subjects

Cheryl E. Praeger

Publications and source records attributed to Cheryl E. Praeger.

At least 37 records · Page 2Linked to original sources

arXiv2024

Higher-dimensional grid-imprimitive block-transitive designs

It was shown in 1989 by Delandtsheer and Doyen that, for a $2$-design with $v$ points and block size $k$, a block-transitive group of automorphisms can be point-imprimitive (that is, leave invariant a nontrivial partition of the point set) only if $v$ is small enough relative to $k$. Recently, exploiting a construction of block-transitive point-imprimitive $2$-designs given by Cameron and the last author, four of the authors studied $2$-designs admitting a block-transitive group that preserves a two-dimensional grid structure on the point set. Here we consider the case where there a block-transitive group preserves a multidimensional grid structure on points. We provide necessary and sufficient conditions for such $2$-designs to exist in terms of the parameters of the grid, and certain `array parameters' which describe a subset of points (which will be a block of the design). Using this criterion, we construct explicit examples of $2$-designs for grids of dimensions three and four, and pose several open questions.

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arXiv2024

Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups

Determining an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on $s$ is attained for some digraph admitting an almost simple $s$-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that $s\leqslant 2$ in the case where the group is an alternating or symmetric group.

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arXiv2024

Chain-imprimitive, flag-transitive 2-designs

We consider $2$-designs which admit a group of automorphisms that is flag-transitive and leaves invariant a chain of nontrivial point-partitions. We build on our recent work on $2$-designs which are block-transitive but not necessarily flag-transitive. In particular we use the concept of the ``array'' of a point subset with respect to the chain of point-partitions; the array describes the distribution of the points in the subset among the classes of each partition. We obtain necessary and sufficient conditions on the array in order for the subset to be a block of such a design. By explicit construction we show that for any $s \geq 2$, there are infinitely many $2$-designs admitting a flag-transitive group that preserves an invariant chain of point-partitions of length $s$. Moreover an exhaustive computer search, using {\sc Magma}, seeking designs with $e_1e_2e_3$ points (where each $e_i\leq 50$) and a partition chain of length $s=3$, produced $57$ such flag-transitive designs, among which only three designs arise from our construction -- so there is still much to learn.

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arXiv2024

Basic Tetravalent Oriented Graphs with Cyclic Normal Quotients

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(Γ, G)$ where $Γ$ is finite, 4-valent, connected, and $G$-oriented ($G$-half-arc-transitive). A subfamily of $\mathcal{OG}(4)$ has recently been identified as `basic' in the sense that all graphs in this family are normal covers of at least one basic member. In this paper we provide a description of such basic pairs which have at least one $G$-normal quotient which is isomorphic to a cycle graph. In doing so, we produce many new infinite families of examples and solve several problems posed in the recent literature on this topic. This result completes a research project aiming to provide a description of all basic pairs in $\mathcal{OG}(4)$.

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arXiv2023

On locally $n \times n$ grid graphs

We investigate locally $n \times n$ grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on $n$ vertices. We consider the subclass of these graphs for which each pair of vertices at distance two is joined by sufficiently many paths of length $2$. The number of such paths is known to be at most $2n$ by previous work of Blokhuis and Brouwer. We show that if each distance two pair is joined by at least $n-1$ paths of length $2$ then the diameter is bounded by $O(\log(n))$, while if each pair is joined by at least $2(n-1)$ such paths then the diameter is at most $3$ and we give a tight upper bound on the order of the graphs. We show that graphs meeting this upper bound are distance-regular antipodal covers of complete graphs. We exhibit an infinite family of such graphs which are locally $n \times n$ grid for odd prime powers $n$, and apply these results to locally $5 \times 5$ grid graphs to obtain a classification for the case where either all $μ$-graphs have order at least $8$ or all $μ$-graphs have order $c$ for some constant $c$.

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arXiv2023

Block-transitive 2-designs with a chain of imprimitive partitions

More than $30$ years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive $2$-design, with blocks of size $k$, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of $k$. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive $2$-design. We introduce the notion of an `array' of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the `array' of a point set, relative to a partition chain, for it to be a block of such a design.

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arXiv2023

Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite $2$-arc-transitive graph which is not a Cayley graph

A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition such that the automorphism group of the Cayley graph $\Cay(H,(X\cup Y)\setminus\{1\})$ is equal to $H: A(H,X,Y)$, where $A(H,X,Y)$ is the setwise stabiliser in $\Aut(H)$ of $X\cup Y$. We use this criterion to resolve a questions of Li, Ma and Pan from 2009, by constructing a $2$-arc transitive normal cover of order $2^{53}$ of the complete bipartite graph $\K_{16,16}$ and prove that it is \emph{not} a Cayley graph.

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arXiv2023

A family of $2$-groups and an associated family of semisymmetric, locally $2$-arc-transitive graphs

A mixed dihedral group is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper, for each $n\geq 2$, we construct a mixed dihedral $2$-group $H$ of nilpotency class $3$ and order $2^a$ where $a=(n^3+n^2+4n)/2$, and a corresponding graph $Σ$, which is the clique graph of a Cayley graph of $H$. We prove that $Σ$ is semisymmetric, that is, ${\rm Aut}(Σ)$ acts transitively on the edges, but intransitively on the vertices, of $Σ$. These graphs are the first known semisymmetric graphs constructed from groups that are not $2$-generated (indeed $H$ requires $2n$ generators). Additionally, we prove that $Σ$ is locally $2$-arc-transitive, and is a normal cover of the `basic' locally $2$-arc-transitive graph ${\rm K}_{2^n,2^n}$. As such, the construction of this family of graphs contributes to the investigation of normal covers of prime-power order of basic locally $2$-arc-transitive graphs -- the `local' analogue of a question posed by C.~H.~Li.

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arXiv2022

Rank three innately transitive permutation groups and related $2$-transitive groups

The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups. This paper extends classifications of finite primitive and quasiprimitive groups of rank at most $3$ to a classification for the finite innately transitive groups. The new examples comprise three infinite families and three sporadic examples. A necessary step in this classification was the determination of certain configurations in finite almost simple $2$-transitive groups called special pairs.

math.GR↗

arXiv2022

A characterisation of edge-affine $2$-arc-transitive covers of $\K_{2^n,2^n}$

We introduce the notion of an \emph{$n$-dimensional mixed dihedral group}, a general class of groups for which we give a graph theoretic characterisation. In particular, if $H$ is an $n$-dimensional mixed dihedral group then the we construct an edge-transitive Cayley graph $Γ$ of $H$ such that the clique graph $Σ$ of $Γ$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(Σ)$ inducing a particular \emph{edge-affine} action on $\K_{2^n,2^n}$. Conversely, we prove that if $Σ$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(Σ)$ inducing an \emph{edge-affine} action on $\K_{2^n,2^n}$, then the line graph $Γ$ of $Σ$ is a Cayley graph of an $n$-dimensional mixed dihedral group. Furthermore, we give an explicit construction of a family of $n$-dimensional mixed dihedral groups. This family addresses a problem proposed by Li concerning normal covers of prime power order of the `basic' $2$-arc-transitive graphs. In particular, we construct, for each $n\geq 2$, a $2$-arc-transitive normal cover of $2$-power order of the `basic' graph $\K_{2^n,2^n}$.

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arXiv2022

Tournaments and Even Graphs are Equinumerous

A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise. Pontus von Brömssen (né Andersson) showed that the existence of such an automorphism is independent of the orientation, and considered the question of counting pairwise non-isomorphic even graphs. Based on computational evidence, he made the rather surprising conjecture that the number of pairwise non-isomorphic even graphs on $n$ vertices is equal to the number of pairwise non-isomorphic tournaments on $n$ vertices. We prove this conjecture using a counting argument with several applications of the Cauchy-Frobenius Theorem.

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arXiv2022

On the proportion of elements of prime order in finite symmetric groups

We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order $p$, acting on a finite set of given size $n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv k\pmod{p}$ with $0\leq k\leq p-1$, then this proportion is at most $(p\cdot k!)^{-1}$ with equality if and only if $p\leq n<2n$.

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arXiv2022

Analysing flag-transitive point-imprimitive 2-designs

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 $4$.

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arXiv2022

Random generation of direct sums of finite non-degenerate subspaces

Let $V$ be a $d$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose $|\mathbb{F}|=q^2$ if $V$ is hermitian, and $|\mathbb{F}|=q$ otherwise. Given integers $e, e'$ such that $e+e'\leqslant d$, we estimate the proportion of pairs $(U, U')$, where $U$ is a non-degenerate $e$-subspace of $V$ and $U'$ is a non-degenerate $e'$-subspace of $V$, such that $U\cap U'=0$ and $U\oplus U'$ is non-degenerate (the sum $U\oplus U'$ is direct and usually not perpendicular). The proportion is shown to be positive and at least $1-c/q>0$ for some constant $c$. For example, $c=7/4$ suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that $\dim(U)$ and $\dim(U')$ are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups $G$ relies on a connection between certain pairs $(U,U')$ of non-degenerate subspaces and certain pairs $(g,g')\in G^2$ of group elements where $U={\rm im}(g-1)$ and $U'={\rm im}(g'-1)$.

math.GR↗

arXiv2022

The probability of spanning a classical space by two non-degenerate subspaces of complementary dimension

Let $n,n'$ be positive integers and let $V$ be an $(n+n')$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs $(U, U')$, where $U$ is a non-degenerate $n$-subspace and $U'$ is a non-degenerate $n'$-subspace of $V$, such that $U+ U'=V$ (usually such spaces $U$ and $U'$ are not perpendicular). The proportion is shown to be at least $1-c/|\mathbb{F}|$ for some constant $c\leqslant 2$ in the symplectic or unitary cases, and $c<3$ in the orthogonal case.

math.GR↗

arXiv2022

Locally Finite Vertex-Rotary Maps and Coset Graphs with Finite Valency and Finite Edge Multiplicity

It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs $\Cos(G,H,J)$ with finite valency and finite edge-multiplicity, where $H, J$ are stabilisers in $G$ of a vertex and incident edge, respectively. Given a group $G=ła,z\r$ with $|z|=2$ and $|a|$ finite, the coset graph $\Cos(G,ła\r,łz\r)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a $G$-arc-transitive map $(V,E,F)$, namely, a {\it $G$-rotary} map if $|az|$ is finite, and a {\it $G$-bi-rotary} map if $|zz^a|$ is finite. The $G$-rotary map can be represented as a coset geometry for $G$, extending the notion of a coset graph. However the $G$-bi-rotary map does not have such a representation, and the face boundary cycles must be specified in addition to incidences between faces and edges. We also give a coset geometry construction of a flag-regular map $(V,E,F)$. In all of these constructions we prove that the face boundary cycles are regular cycles which are simple cycles precisely when the given group acts faithfully on $V\cup F$.

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arXiv2022

Codes and Designs in Johnson Graphs From Symplectic Actions on Quadratic Forms

The Johnson graph $J(v, k)$ has as vertices the $k$-subsets of $\mathcal{V}=\{1,\ldots, v\}$, and two vertices are joined by an edge if their intersection has size $k-1$. An \emph{$X$-strongly incidence-transitive code} in $J (v, k)$ is a proper vertex subset $Γ$ such that the subgroup $X$ of graph automorphisms leaving $Γ$ invariant is transitive on the set $Γ$ of `codewords', and for each codeword $Δ$, the setwise stabiliser $X_Δ$ is transitive on $Δ\times (\mathcal{V}\setminus Δ)$. We classify the \emph{$X$-strongly incidence-transitive codes} in $J(v,k)$ for which $X$ is the symplectic group $\mathrm{Sp}_{2n}(2)$ acting as a $2$-transitive permutation group of degree $2^{2n-1}\pm 2^{n-1}$, where the stabiliser $X_Δ$ of a codeword $Δ$ is contained in a \emph{geometric} maximal subgroup of $X$. In particular, we construct two new infinite families of strongly incidence-transitive codes associated with the reducible maximal subgroups of $\mathrm{Sp}_{2n}(2)$.

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arXiv2022

Block-transitive two-designs based on grids

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-interpret these conditions more graph theoretically, and then focus on square grids, and designs admitting the full automorphism group of $\mathbf{K}_{m,m}$. We find necessary and sufficient conditions, again in terms of graph theoretic parameters, for these incidence structures to be $t$-designs, for $t=2, 3$, and give infinite families of examples illustrating that block-transitive, point-primitive $2$-designs based on grids exist for all values of $m$, and flag-transitive, point-primitive examples occur for all even $m$. This approach also allows us to construct a small number of block-transitive $3$-designs based on grids.

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