arXiv subjects

Cheryl E. Praeger

Publications and source records attributed to Cheryl E. Praeger.

At least 55 records · Page 3Linked to original sources

arXiv2021

Totally $2$-closed finite groups with trivial Fitting subgroup

A group $G$ is said to be totally $2$-closed if in each of its faithful permutation representations, say on a set $Ω$, $G$ is the largest subgroup of $\mathrm{Sym}(Ω)$ which leaves invariant each of the $G$-orbits for the induced action on $Ω\times Ω$. We prove that there are precisely $47$ finite totally $2$-closed groups with trivial Fitting subgroup. Each of these groups is a direct product of pairwise non-isomorphic sporadic simple groups, with the direct factors coming from the Janko groups $\mathrm{J}_1, \mathrm{J}_3$ and $\mathrm{J}_4$, together with $\mathrm{Ly}, \mathrm{Th}$ and the Monster $\mathbb{M}$. These are the first known examples of insoluble totally $2$-closed groups. As a by-product of our methods, we develop several tools for studying $2$-closures of transitive permutation groups -- a vital tool in the study of representations of finite groups as automorphism groups of digraphs. We also prove a dual to a 1939 theorem of Frucht from Algebraic Graph Theory.

math.GR↗

arXiv2021

The graphs with a symmetrical Euler cycle

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 symmetrical Euler cycle of $[C]$. In this paper we determine the family of graphs (which may have multiple edges) whose edge-sets can be sequenced to form a symmetrical Euler cycle. We first classify all graphs which have a cyclic subgroup of automorphisms acting bi-regularly on edges. We then apply this classification to obtain the graphs possessing a symmetrical Euler cycle, and therefore are the (only) candidates for the induced subgraphs of the boundary cycles of the faces of arc-transitive maps.

math.CO↗

arXiv2021

Sequences of linear codes where the rate times distance grows rapidly

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 $[n_i, k_i, d_i]$ such that $k_id_i/n_i$ grows quickly in the sense that $k_id_i/n_i>\sqrt{k_i}-1>2i-1$. Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is $\mathbb{F}=\mathbb{F}_2$ and $k_i d_i/n_i$ is asymptotic to $3n_i^{c}/\sqrt{π\log_2(n_i)}$ where $c=\log_2(3/2)\approx 0.585$.

cs.IT↗

arXiv2021

The geometry of diagonal groups

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 been studied. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over any group, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our axioms. However, for m>=3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions are equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group except in four small cases with m<=3. The class of diagonal graphs includes some well-known families, Latin-square graphs and folded cubes. We obtain partial results on the chromatic number of a diagonal graph, and mention an application to synchronization.

math.GR↗

arXiv2021

Most permutations power to a cycle of small prime length

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 property is at least $1-7/\log\log n$.

math.GR↗

arXiv2021

Diagonal groups and arcs over groups

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 $Ω$, 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 over a base group $G$. In this paper we investigate what happens if we have $m+r$ partitions with $r\geq 2$, any $m$ of which are minimal elements of a Cartesian lattice. If $m=2$, this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For $m>2$, things are more restricted. Any $m+1$ of the partitions generate a join-semilattice admitting a diagonal group over a group $G$. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that $G$ must be abelian and must have three fixed-point-free automorphisms whose product is the identity. Under this hypothesis, such a structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order $p$, then the structure corresponds exactly to an arc of cardinality $m+r$ in the $(m-1)$-dimensional projective space over the field with $p$ elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order $q$ give examples where $G$ is the elementary abelian group of order $q$. These examples can be lifted to non-elementary abelian groups using $p$-adic techniques.

math.CO↗

arXiv2021

Partial linear spaces with a rank 3 affine primitive group of automorphisms

A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on points. In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of $AΓL_1(q)$. Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups. We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest.

math.GR↗

arXiv2021

Normal edge-transitive Cayley graphs and Frattini-like subgroups

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 $Aut(G;C)$ acts transitively on the pairs $\{c,c^{-1}\}$ from $C$. We show that, for a normal edge-transitive Cayley graph $Cay(G,C)$, its quotient modulo $Φ(G;C)$ is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever the subgroup $Φ(G;C)$ is trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all $4$-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi.

math.GR↗

arXiv2021

Finite totally $k$-closed groups

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 $Ω$, $G$ is the largest subgroup of $\operatorname{Sym}(Ω)$ which leaves invariant each of the $G$-orbits in the induced action on $Ω\times\dots\times Ω=Ω^k$. We prove that every abelian group $G$ is totally $(n(G)+1)$-closed, but is not totally $n(G)$-closed, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$. In particular, we prove that for each $k\geq2$ and each prime $p$, there are infinitely many finite abelian $p$-groups which are totally $k$-closed but not totally $(k-1)$-closed. This result in the special case $k=2$ is due to Abdollahi and Arezoomand. We pose several open questions about total $k$-closure.

math.GR↗

arXiv2021

Orbits of Sylow subgroups of finite permutation groups

We say that a finite group $G$ acting on a set $Ω$ has Property $(*)_p$ for a prime $p$ if $P_ω$ is a Sylow $p$-subgroup of $G_ω$ for all $ω\inΩ$ 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 $A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(*)_p$. In this paper, we extend this result to finite $2$-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy Property $(*)_p$ for an allowable prime $p$.

math.GR↗

arXiv2020

Subgroups of Classical Groups that are Transitive on Subspaces

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 classify all point-transitive subgroups of automorphisms of finite thick generalised quadrangles.

math.GR↗

arXiv2020

Point-primitive generalised hexagons and octagons and projective linear groups

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 $\mathcal{S}$ is a finite thick generalised hexagon or octagon with $G \leqslant{\rm Aut}(\mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${\rm PSL}_n(q)$ where $n \geqslant 2$, then the stabiliser of a point acts irreducibly on the natural module. We describe a strategy to prove that such a generalised hexagon or octagon $\mathcal{S}$ does not exist.

math.GR↗

arXiv2020

On flag-transitive imprimitive 2-designs

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 $100$ points. There are exactly eleven such examples, and for one of these designs, a flag-regular, point-imprimitive $2-(36,8,4)$ design with automorphism group ${\rm Sym}(6)$, there seems to be no construction previously available in the literature.

math.CO↗

arXiv2020

Automorphism orbits and element orders in finite groups: almost-solubility and the Monster

For a finite group $G$, we denote by $ω(G)$ the number of $\operatorname{Aut}(G)$-orbits on $G$, and by $\operatorname{o}(G)$ the number of distinct element orders in $G$. In this paper, we are primarily concerned with the two quantities $\mathfrak{d}(G):=ω(G)-\operatorname{o}(G)$ and $\mathfrak{q}(G):=ω(G)/\operatorname{o}(G)$, each of which may be viewed as a measure for how far $G$ is from being an AT-group in the sense of Zhang (that is, a group with $ω(G)=\operatorname{o}(G)$). We show that the index $|G:\operatorname{Rad}(G)|$ of the soluble radical $\operatorname{Rad}(G)$ of $G$ can be bounded from above both by a function in $\mathfrak{d}(G)$ and by a function in $\mathfrak{q}(G)$ and $\operatorname{o}(\operatorname{Rad}(G))$. We also obtain a curious quantitative characterisation of the Fischer-Griess Monster group $\operatorname{M}$.

math.GR↗

arXiv2020

Delandtsheer--Doyen parameters for block-transitive point-imprimitive 2-designs

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 designs achieving these bounds, and pose several open questions concerning the Delandtsheer--Doyen parameters.

math.CO↗

arXiv2020

Generating infinite digraphs by derangements

A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set $V$ generates a digraph with vertex set $V$ and arcs $(x,x^σ)$ for $x\in V$ and $σ\in\mathcal{S}$. We address the problem of characterising those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was addressed in earlier work by the second and third authors. A criterion is given for derangement generation which resembles the criterion given by De Bruijn and Erdős for vertex colourings of graphs in that the property for an infinite digraph is determined by properties of its finite sub-digraphs. The derangement generation property for a digraph is linked with the existence of a finite $1$-factor cover for an associated bipartite (undirected) graph.

math.CO↗

arXiv2020

A path-deformation framework for determining weighted genome rearrangement distance

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 to use the group-theoretic framework to establish minimal distance for any weighting on the set of inversions, generalizing previous approaches. To do this we use the theory of rewriting systems for groups, and exploit the Knuth--Bendix algorithm, the first time this theory has been introduced into genome rearrangement problems. The central idea of the approach is to use existing group theoretic methods to find an initial path between two genomes in genome space (for instance using only short inversions), and then to deform this path to optimality using a confluent system of rewriting rules generated by the Knuth--Bendix algorithm.

math.CO↗

arXiv2020

Conjugacy class sizes in arithmetic progression

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 with $r\geqslant 2$. (We show that ${\rm cs}(G)=\{1,2,3\}$.) Our most substantial result classifies all $G$ with ${\rm cs}^*(G)=\{2,4,6\}$. Finally, we classify all groups $G$ whose largest two non-central conjugacy class sizes are coprime. (Here it is not obvious but it is true that ${\rm cs}^*(G)$ has two elements, and so is an arithmetic progression.)

math.GR↗