Cheryl E. Praeger

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• D Phil, DSc
• Professor Emeritus at The University of Western Australia

A partial linear space is a point--line incidence structure such that each line is incident with at least two points and each pair of points is incident with at most one line. It is said to be proper if there exists at least one non-collinear point pair, and at least one line incident with more than two points. The highest degree of symmetry for a...

This chapter provides an overview of codes in distance regular graphs which admit symmetries via a permutation group acting on the vertices of the graph. The strongest notion of completely transitive codes is developed, as well as the more general notion of neighbour transitive codes. The graphs considered are the Hamming, Johnson, and Kneser graph...

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...

Let V := (Fq)^d be a d-dimensional vector space over the field Fq of order q. Fix positive integers e, e' satisfying e+e' = d. Motivated by analysing a fundamental algorithm in computational group theory for recognising classical groups, we consider a certain quantity P(e,e') which arises in both graph theory and group representation theory: P(e,e'...

For a finite group $A$ with normal subgroup $G$, a subgroup $U$ of $G$ is an $A$-prime-power-covering subgroup if $U$ meets every $A$-conjugacy-class of elements of $G$ of prime power order. It is conjectured that $|G:U|$ is bounded by some function of $|A:G|$, and this conjecture has number theoretic implications for relative Brauer groups of alge...

For an element $x$ of a finite group $T$, the $\mathrm{Aut}(T)$-class of $x$ is the set $\{ x^\sigma\mid \sigma\in \mathrm{Aut}(T)\}$. We prove that the order $|T|$ of a finite nonabelian simple group $T$ is bounded above by a function of the parameter $m(T)$, where $m(T)$ is the maximum, over all primes $p$, of the number of $\mathrm{Aut}(T)$-clas...

This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $GL_d(\mathbb{F}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module $V=(\mathbb{F}_q)^d$, is irreducible on a subspace of the form $V(1-g)$ of dimension $d/2$. We classify $G,d,r$, the characteristi...

We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory requirements arising during evaluation. We introduce an approach for formalizing this idea and, as illustration,...

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 third author in 1989, existence of a vertex-primitive $2$-arc-transitive digraph was not known until an infinite...

The family $\mathcal{OG}(4)$ consisting of graph-group pairs $(\Gamma, G)$, where $\Gamma$ is a finite, connected, 4-valent graph admitting a $G$-vertex-, and $G$-edge-transitive, but not $G$-arc-transitive action, has recently been examined using a normal quotient methodology. A subfamily of $\mathcal{OG}(4)$ has been identified as `basic', due to...

This is a chapter in a forthcoming book on completely regular codes in distance regular graphs. The chapter provides an overview, and some original results, on codes in distance regular graphs which admit symmetries via a permutation group acting on the vertices of the graph. The strongest notion of completely transitive codes is developed, as well...

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 arc-transitive group $X$ of automorphisms, such that $\Gamma$ has a 2-cell embedding as a map on a closed sur...

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-...

The family OG(4) consisting of graph-group pairs (Γ, G), where Γ is a finite, connected, 4-valent graph admitting a G-vertex-, and G-edge-transitive, but not G-arc-transitive action, has recently been examined using a normal quotient methodology. A subfamily of OG(4) has been identified as ‘basic’, due to the fact that all members of OG(4) are norm...

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, we give a sufficient condition such that the automorphism group of the Cayley graph \(\textrm{Cay}(H,(X\cup Y){\setminus }\{1\})\) is equal to \(H\rti...

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+4...

The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of fi�nite transitive permutation groups, and are analogues of composition factors for abstract fi�nite groups. This paper extends classi�fications of fi�nite primitive and quasiprimitive groups of rank at most 3 to a classi...

For a finite group G G , we denote by ω ( G ) \omega (G) the number of A u t ( G ) Aut(G) -orbits on G G , and by o ( G ) o(G) the number of distinct element orders in G G . In this paper, we are primarily concerned with the two quantities d ( G ) ≔ ω ( G ) − o ( G ) \mathfrak {d}(G)≔\omega (G)-o(G) and q ( G ) ≔ ω ( G ) / o ( G ) \mathfrak {q}(G)≔...

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)...

For a linear code C of length n with dimension k and minimum distance d, it is desirable that the quantity kd/n is large. Given an arbitrary field F, we introduce a novel, but elementary, construction that produces a recursively defined sequence of F-linear codes C1 , C2 , C3, . . . with parameters [n_i, k_i, d_i ] such that k_id_i/n_i grows quickl...

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 f...

For a positive integer k, a group G is said to be totally k-closed if for each set Ω upon which G acts faithfully, G is the largest subgroup of Sym(Ω) that leaves invariant each of the G-orbits in the induced action on Ω×⋯×Ω=Ωk. Each finite group G is totally |G|-closed, and k(G) denotes the least integer k such that G is totally k-closed. We addre...

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 correspondin...

A finite permutation group [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] is the largest subgroup of [Formula: see text] which leaves invariant each of the [Formula: see text]-orbits for the induced action on [Formula: see text]. Introduced by Wielandt in 1969, the concept of [Formula: see text]-closure has develope...

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...

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 classifi...

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 $\Gamma$ of $H$ such that the clique graph $\Sigma$ of $\Gamma$ is a $2$-arc...

We study point‐block incidence structures for which the point set is an grid. Cameron and the fourth author showed that each block may be viewed as a subgraph of a complete bipartite graph with bipartite parts (biparts) of sizes . In the case where consists of all the subgraphs isomorphic to , under automorphisms of fixing the two biparts, they obt...

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...

Let n,n′ be positive integers and let V be an (n+n′)-dimensional vector space over a finite field 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′ ar...

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-impr...

Abstract to Obituary: Peter Neumann was a remarkable man, who made profound and lasting contributions to mathematics and its history. He was passionate about teaching and mathematics education at all levels, and inspired many others. He was a superb lecturer, writer and expositor, and delightful company with a sparkling sense of humour. Above all...

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if for each set $\Omega$ upon which $G$ acts faithfully, $G$ is the largest subgroup of $\mathrm{Sym}(\Omega)$ that leaves invariant each of the $G$-orbits in the induced action on $\Omega\times\cdots\times \Omega=\Omega^k$. Each finite group $G$ is totally $|G|$-closed, and $...

Let V be a d-dimensional vector space over a finite field F equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose |F|=q2 if V is hermitian, and |F|=q otherwise. Given integers e,e′ such that e+e′⩽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′-subsp...

A graph is called \emph{odd} if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and \emph{even} otherwise. Pontus von Br\"omssen (n\'e Andersson) showed that the existence of such an automorphism is independent of the orientation, and considered the question of counting pairwise non-is...

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 , n , now called Delandtsheer–Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsh...

In this article we discuss women’s participation in and contribution to the mathematical sciences in Australia, focusing on efforts to remove barriers to participation, and in particular WIMSIG, the Women in Mathematics Special Interest Group of the Australian Mathematical Society.We are proud of and grateful for what women in Australia have achiev...

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 g...

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 $\Gamma$ such that the subgroup $X$ of graph automorphisms leaving $\Gamma$ invariant is t...

The graphs in this paper are finite, undirected, and without loops, but may have more than one edge between a pair of vertices. If such a graph has ℓ edges, then an Euler cycle is a sequence (e1,e2,…,eℓ) of these ℓ edges, each occurring exactly once, such that ei, ei + 1 are incident with a common vertex for each i (reading subscripts modulo ℓ). An...

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 $\mathcal{B}$ cons...

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-deg...

A group $G$ is said to be totally $2$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\mathrm{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits for the induced action on $\Omega\times \Omega$. We prove that there are precisely $47$ finite totally $2$-closed groups with...

The edges surrounding a face of a map $M$ form a cycle $C$, called the boundary cycle of the face, and $C$ is often not a simple cycle. If the map $M$ is arc-transitive, then there is a cyclic subgroup of automorphisms of $M$ which leaves $C$ invariant and is bi-regular on the edges of the induced subgraph $[C]$; that is to say, $C$ is a symmetrica...

For a linear code $C$ of length $n$ with dimension $k$ and minimum distance $d$, it is desirable that the quantity $kd/n$ is large. Given an arbitrary field $\mathbb{F}$, we introduce a novel, but elementary, construction that produces a recursively defined sequence of $\mathbb{F}$-linear codes $C_1,C_2, C_3, \dots$ with parameters $[n_i, k_i, d_i]...

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 $U+ U'=V$...

The investigation of s-arc-transitivity of digraphs can be dated back to 1989 when the third author showed that s can be arbitrarily large if the action on vertices is imprimitive. However, the situation is completely different when the digraph is vertex-primitive and not a directed cycle. In 2017 the second author, Li and Xia constructed the first...

Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an {\em association scheme on triples} (AST) and constructed...

The mathematics of shuffling a deck of 2n cards with two “perfect shuffles” was brought into clarity by Diaconis, Graham and Kantor. Here we consider a generalisation of this problem, with a so-called “many handed dealer” shuffling kn cards by cutting into k piles with n cards in each pile and using k! shuffles. A conjecture of Medvedoff and Morris...

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $$m\ge 2$$ m ≥ 2 , a set of $$m+1$$ m + 1 partitions of a set $$\Omega $$ Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $$m=2$$ m = 2 ), or generate a join-semilattice of dimension m associ...

Diagonal groups are one of the classes of �finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have no...

This volume contains nine survey articles based on plenary lectures given at the 28th British Combinatorial Conference, hosted online by Durham University in July 2021. This biennial conference is a well-established international event, attracting speakers from around the world. Written by some of the foremost researchers in the field, these survey...

We say that a finite group G acting on a set Ω has Property (⁎)p for a prime p if Pω is a Sylow p-subgroup of Gω for all ω∈Ω and Sylow p-subgroups P of G. Property (⁎)p arose in the recent work of Tornier (2018) on local Sylow p-subgroups of Burger-Mozes groups, and he determined the values of p for which the alternating group An and symmetric grou...

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$ . Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$ . The proportion of even permutations with this proper...

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 ⩽ 4 we obtain a list of “numerically feasible”...

We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if S is a f...

Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an association scheme on triples (AST) and constructed exampl...

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 rema...

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 $\Phi(G;C)$ 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...

An s-geodesic of a graph is a path of length s such that the first and last vertices are at distance s. We study finite graphs Γ of diameter at least 3 for which some subgroup G of automorphisms is transitive on the set of s-geodesics for each s≤3. If Γ has girth at least 6 then all 3-arcs are 3-geodesics so Γ is 3-arc-transitive, and such graphs h...

We say that a finite group $G$ acting on a set $\Omega$ has Property $(*)_p$ for a prime $p$ if $P_\omega$ is a Sylow $p$-subgroup of $G_\omega$ for all $\omega\in\Omega$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes groups, and he determined the values o...

This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag trans...

MATRIX is Australia’s international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the ten programs held at MATRIX in 2019 and the two programs held in January 2020: · Topolo...

For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the classification of the maximal factorisations of almost simple groups. As a first application of these results we...

We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if $\mathca...

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits in the induced action on $\Omega\times\dots\times \Omega=\Omega^k$. We prove that every abel...

A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set V generates a digraph with vertex set V and arcs $(x,x^{\,\sigma})$ for x ∈ V and $\sigma\in\mathcal{S}$. We address the problem of characterizing those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was...

In this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may...

In this paper, we first study biplanes \(\mathcal {D}\) with parameters (v, k, 2), where the block size \(k\in \{13,16\}\). These are the smallest parameter values for which a classification is not available. We show that if \(k=13\), then either \(\mathcal {D}\) is the Aschbacher biplane or its dual, or \(\mathbf {Aut}(\mathcal {D})\) is a subgrou...

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $m\ge2$, a set of $m+1$ partitions of a set $\Omega$, any $m$ of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $m=2$), or generate a join-semilattice of dimension $m$ associated with a diagonal group ov...

Measuring the distance between two bacterial genomes under the inversion process is usually done by assuming all inversions to occur with equal probability. Recently, an approach to calculating inversion distance using group theory was introduced, and is effective for the model in which only very short inversions occur. In this paper, we show how t...

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 Deland...

Measuring the distance between two bacterial genomes under the inversion process is usually done by assuming all inversions to occur with equal probability. Recently, an approach to calculating inversion distance using group theory was introduced, and is effective for the model in which only very short inversions occur. In this paper, we show how t...

Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not...

In 1987, Huw Davies proved that, for a flag-transitive point-imprimitive $2$-$(v,k,\lambda)$ design, both the block-size $k$ and the number $v$ of points are bounded by functions of $\lambda$, but he did not make these bounds explicit. In this paper we derive explicit polynomial functions of $\lambda$ bounding $k$ and $v$. For $\lambda\leq 4$ we ob...

A graph Γ is k-connected-homogeneous (k-CH) if k is a positive integer and any isomorphism between connected induced subgraphs of order at most k extends to an automorphism of Γ, and connected-homogeneous (CH) if this property holds for all k. Locally finite, locally connected graphs often fail to be 4-CH because of a combinatorial obstruction call...

Let {\mathrm{cs}(G)} denote the set of conjugacy class sizes of a group G , and let \mathrm{cs}^{*}(G)=\mathrm{cs}(G)\setminus\{1\} be the sizes of non-central classes. We prove three results. We classify all finite groups for which (1) {\mathrm{cs}(G)=\{a,a+d,\dots,a+rd\}} is an arithmetic progression with {r\geqslant 2} ; (2) {\mathrm{cs}^{*}(G)=...

In this paper, we first study biplanes $\mathcal{D}$ with parameters $(v,k,2)$, where the block size $k\in\{13,16\}$. These are the smallest parameter values for which a classification is not available. We show that if $k=13$, then either $\mathcal{D}$ is the Aschbacher biplane or its dual, or $Aut(\mathcal{D})$ is a subgroup of the cyclic group of...

Let ${\rm cs}(G)$ denote the set of conjugacy class sizes of a group $G$, and let ${\rm cs}^*(G)={\rm cs}(G)\setminus\{1\}$ be the sizes of non-central classes. We prove three results. We classify all finite groups $G$ with ${\rm cs}(G)=\{a, a+d, \dots ,a+rd\}$ an arithmetic progression involving at most two primes and $r\geqslant 2$. Our most diff...

The normal covering number γ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (G)$$\end{document} of a finite, non-cyclic group G is the minimum number of prope...

A code $C$ in the Hamming graph $\varGamma=H(m,q)$ is $2$-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...

This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag trans...

Let $m$ be a positive integer and let $\rho(m,n)$ be the proportion of permutations of the symmetric group $\mathrm{Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\rho(n,m)n^{1-\frac{\phi(m)}{m}}\sim \kappa_m$ where $\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$...

MATRIX is Australia’s international and residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each 1-4 weeks in duration. This book is a scientific record of the eight programs held at MATRIX in 2018: - Non-Equilibrium Systems and Special Functions -...

We prove that most permutations of degree $n$ power to a cycle of prime length approximately $\log n$. Explicitly, we show that for $n> 163\,000$, the proportion of such elements is at least $1-5/\log\log n$ with the prime between $\log n$ and $(\log n)^{\log\log n}$. For even permutations this proportion is at least $1-7/\log\log n$.

The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum distance at least 5 via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained her...