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Cheryl E. Praeger

Publications and source records attributed to Cheryl E. Praeger.

At least 163 records · Page 9Linked to original sources
arXiv2011

Symmetry properties of subdivision graphs

The subdivision graph $S(Σ)$ of a graph $Σ$ is obtained from $Σ$ by `adding a vertex' in the middle of every edge of $\Si$. Various symmetry properties of $§(Σ)$ are studied. We prove that, for a connected graph $Σ$, $S(Σ)$ is locally $s$-arc transitive if and only if $Σ$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. The diameter of $S(Σ)$ is $2d+δ$, where $Σ$ has diameter $d$ and $0\leqslant δ\leqslant 2$, and local $s$-distance transitivity of $§(Σ)$ is defined for $1\leqslant s\leqslant 2d+δ$. In the general case where $s\leqslant 2d-1$ we prove that $S(Σ)$ is locally $s$-distance transitive if and only if $Σ$ is $\lceil\frac{s+1}{2}\rceil$-arc transitive. For the remaining values of $s$, namely $2d\leqslant s\leqslant 2d+δ$, we classify the graphs $Σ$ for which $S(Σ)$ is locally $s$-distance transitive in the cases, $s\leqslant 5$ and $s\geqslant 15+δ$. The cases $\max\{2d, 6\}\leqslant s\leqslant \min\{2d+δ, 14+δ\}$ remain open.

math.GR↗
arXiv2010

Set-homogeneous directed graphs

A directed graph is set-homogeneous if, whenever U and V are isomorphic finite subdigraphs, there is an automorphism g of the digraph with U^g=V. Here, extending work of Lachlan on finite homogeneous digraphs, we classify finite set-homogeneous digraphs, where we allow some pairs of vertices to have arcs in both directions. Under the assumption that such pairs of vertices are not allowed, we obtain initial results on countably infinite set-homogeneous digraphs, classifying those which are not 2-homogeneous.

math.CO↗
arXiv2010

Locally $s$-distance transitive graphs

We give a unified approach to analysing, for each positive integer $s$, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally $s$-arc transitive graphs of diameter at least $s$. A graph is in the class if it is connected and if, for each vertex $v$, the subgroup of automorphisms fixing $v$ acts transitively on the set of vertices at distance $i$ from $v$, for each $i$ from 1 to $s$. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for $s\geq 2$, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.

math.CO↗
arXiv2010

Basic and degenerate pregeometries

We study pairs $(Γ,G)$, where $Γ$ is a 'Buekenhout-Tits' pregeometry with all rank 2 truncations connected, and $G\leqslant\mathrm{Aut} Γ$ is transitive on the set of elements of each type. The family of such pairs is closed under forming quotients with respect to $G$-invariant type-refining partitions of the element set of $Γ$. We identify the 'basic' pairs (those that admit no non-degenerate quotients), and show, by studying quotients and direct decompositions, that the study of basic pregeometries reduces to examining those where the group $G$ is faithful and primitive on the set of elements of each type. We also study the special case of normal quotients, where we take quotients with respect to the orbits of a normal subgroup of $G$. There is a similar reduction for normal-basic pregeometries to those where $G$ is faithful and quasiprimitive on the set of elements of each type.

math.CO↗
arXiv2010

A new solvability criterion for finite groups

In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition: $G$ is solvable if for all conjugacy classes $C$ and $D$ of $G$, \emph{there exist} $x\in C$ and $y\in D$ for which $\gen{x,y}$ is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if $G$ is a finite nonabelian simple group, then there exist two integers $a$ and $b$ which represent orders of elements in $G$ and for all elements $x,y\in G$ with $|x|=a$ and $|y|=b$, the subgroup $\gen{x,y}$ is nonsolvable.

math.GR↗
arXiv2010

Constructive membership testing in black-box classical groups

The research described in this note aims at solving the constructive membership problem for the class of quasisimple classical groups. Our algorithms are developed in the black-box group model; that is, they do not require specific characteristics of the representations in which the input groups are given. The elements of a black-box group are represented, not necessarily uniquely, as bit strings of uniform length. We assume the existence of oracles to compute the product of two elements, the inverse of an element, and to test if two strings represent the same element. Solving the constructive membership problem for a black-box group $G$ requires to write every element of $G$ as a word in a given generating set. In practice we write the elements of $G$ as straight-line programs (SLPs) which can be viewed as a compact way of writing words.

math.GR↗
arXiv2009

Bipartite divisor graphs for integer subsets

Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph $\Ga(X)$ and the prime vertex graph $Δ(X)$, for a set $X$ of positive integers, we define the bipartite divisor graph $B(X)$, and show that many of these connections flow naturally from properties of $B(X)$. In particular we establish links between parameters of these three graphs, such as number and diameter of components, and we characterise bipartite graphs that can arise as $B(X)$ for some $X$. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of $B(X)$, for one $Γ(X)$ or $Δ(X)$ to contain a complete subgraph of size 3 or 4.

math.CO↗
arXiv2009

On triple factorisations of finite groups

This paper introduces and develops a general framework for studying triple factorisations of the form $G=ABA$ of finite groups $G$, with $A$ and $B$ subgroups of $G$. We call such a factorisation nondegenerate if $G\neq AB$. Consideration of the action of $G$ by right multiplication on the right cosets of $B$ leads to a nontrivial upper bound for $|G|$ by applying results about subsets of restricted movement. For $A<C<G$ and $B<D<G$ the factorisation $G=CDC$ may be degenerate even if $G=ABA$ is nondegenerate. Similarly forming quotients may lead to degenerate triple factorisations. A rationale is given for reducing the study of nondegenerate triple factorisations to those in which $G$ acts faithfully and primitively on the cosets of $A$. This involves study of a wreath product construction for triple factorisations.

math.GR↗
arXiv2009

Strongly regular edge-transitive graphs

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 the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parameters of the graphs in this family that reduce to complete graphs.

math.CO↗
arXiv2009

On generalised Paley graphs and their automorphism groups

The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the parameters for this to occur. We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in \cite{LLP} to distinguish between cyclotomic schemes and similar twisted versions of these schemes, in the context of homogeneous factorisations of complete graphs.

math.CO↗
arXiv2008

Fundamental Dominations in Graphs

Nine variations of the concept of domination in a simple graph are identified as fundamental domination concepts, and a unified approach is introduced for studying them. For each variation, the minimum cardinality of a subset of dominating elements is the corresponding fundamental domination number. It is observed that, for each nontrivial connected graph, at most five of these nine numbers can be different, and inequalities between these five numbers are given. Finally, these fundamental dominations are interpreted in terms of the total graph of the given graph, a concept introduced by the second author in 1965. It is argued that the very first domination concept, defined by O. Ore in 1962 and under a different name by C. Berge in 1958, deserves to be called the most fundamental of graph dominations.

math.CO↗
arXiv2008

Computing Minimal Polynomials of Matrices

We present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of an $n\times n$ matrix over a finite field that requires $O(n^3)$ field operations and O(n) random vectors, and is well suited for successful practical implementation. The algorithm, and its complexity analysis, use standard algorithms for polynomial and matrix operations. We compare features of the algorithm with several other algorithms in the literature. In addition we present a deterministic verification procedure which is similarly efficient in most cases but has a worst-case complexity of $O(n^4)$. Finally, we report the results of practical experiments with an implementation of our algorithms in comparison with the current algorithms in the {\sf GAP} library.

math.RA↗
arXiv2007

Primitive decompositions of Johnson graphs

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.

math.CO↗
arXiv2007

Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight

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 algorithms that, for gcd(k,r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k,r) at most 8, and complete the classification in this case.

math.CO↗
arXiv2006

On the frequency of permutations containing a long cycle

A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the conditional probabilities that an element of $S_n$ or $A_n$ contains an $r$-cycle, given that it satisfies an equation of the form $x^{rs}=1$ where $s\leq3$. For example, the conditional probability that an element $x$ is an $n$-cycle, given that $x^n=1$, is always greater than 2/7, and is greater than 1/2 if $n$ does not divide 24. Our results improve estimates of these conditional probabilities in earlier work of the authors with Beals, Leedham-Green and Seress, and have applications for analysing black-box recognition algorithms for the finite symmetric and alternating groups.

math.GR↗
arXiv2005

Generalised sifting in black-box groups

We present a generalisation of the sifting procedure introduced originally by Sims for computation with finite permutation groups, and now used for many computational procedures for groups, such as membership testing and finding group orders. Our procedure is a Monte Carlo algorithm, and is presented and analysed in the context of black-box groups. It is based on a chain of subsets instead of a subgroup chain. Two general versions of the procedure are worked out in detail, and applications are given for membership tests for several of the sporadic simple groups. Our major objective was that the procedures could be proved to be Monte Carlo algorithms, and their costs computed. In addition we explicitly determined suitable subset chains for six of the sporadic groups, and we implemented the algorithms involving these chains in the {\sf GAP} computational algebra system. It turns out that sample implementations perform well in practice. The implementations will be made available publicly in the form of a {\sf GAP} package.

math.GR↗
arXiv2004

Quasiprimitive groups and blow-up decompositions

The blow-up construction by L. G. Kovács has been a very useful tool to study embeddings of finite primitive permutation groups into wreath products in product action. In the present paper we extend the concept of a blow-up to finite quasiprimitive permutation groups, and use it to study embeddings of finite quasiprimitive groups into wreath products.

math.GR↗