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Peter J. Cameron

Publications and source records attributed to Peter J. Cameron.

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arXiv2020

Between the enhanced power graph and the commuting graph

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting graph of $G$. Two elements of $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that the automorphism group of $G$ acts as automorphisms of the deep commuting graph.

math.CO↗
arXiv2020

Trees and cycles

Let $T$ be a tree on $n$ vertices. We can regard the edges of $T$ as transpositions of the vertex set; their product (in any order) is a cyclic permutation. All possible cyclic permutations arise (each exactly once) if and only if the tree is a star. In this paper we find the number of realised cycles, and obtain some results on the number of realisations of each cycle, for other trees. We also solve the inverse problem of the number of trees which give rise to a given cycle. On the way, we meet some familiar number sequences including the Euler and Fuss--Catalan numbers.

math.CO↗
arXiv2020

Association schemes for diagonal groups

For any finite group $G$, and any positive integer $n$, we construct an association scheme which admits the diagonal group $D_n(G)$ as a group of automorphisms. The rank of the association scheme is the number of partitions of $n$ into at most $|G|$ parts, so is $p(n)$ if $|G|\ge n$; its parameters depend only on $n$ and $|G|$. For $n=2$, the association scheme is trivial, while for $n=3$ its relations are the Latin square graph associated with the Cayley table of $G$ and its complement. A transitive permutation group $G$ is said to be \emph{AS-free} if there is no non-trivial association scheme admitting $G$ as a group of automorphisms. A consequence of our construction is that an AS-free group must be either $2$-homogeneous or almost simple. We construct another association scheme, finer than the above scheme if $n>3$, from the Latin hypercube consisting of $n$-tuples of elements of $G$ with product the identity.

math.GR↗
arXiv2020

The non-commuting, non-generating graph of a nilpotent group

For a nilpotent group $G$, let $Ξ(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $Ξ(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $Ξ^+(G)$ be the subgraph of $Ξ(G)$ induced by its non-isolated vertices. We show that if $Ξ(G)$ has an edge, then $Ξ^+(G)$ is connected with diameter $2$ or $3$, with $Ξ(G) = Ξ^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $Ξ(G)$ in more detail.

math.GR↗
arXiv2020

Groups generated by derangements

We examine the subgroup $D(G)$ of a transitive permutation group $G$ which is generated by the derangements in $G$. Our main results bound the index of this subgroup: we conjecture that, if $G$ has degree $n$ and is not a Frobenius group, then $|G:D(G)|\leqslant\sqrt{n}-1$; we prove this except when $G$ is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding $|H:R(H)|$, where $H$ is a linear group on a finite vector space and $R(H)$ is the subgroup of $H$ generated by elements having eigenvalue~$1$. If $G$ is a Frobenius group, then $D(G)$ is the Frobenius kernel, and so $G/D(G)$ is isomorphic to a Frobenius complement. We give some examples where $D(G)\ne G$, and examine the group-theoretic structure of $G/D(G)$; in particular, we construct groups $G$ in which $G/D(G)$ is not a Frobenius complement.

math.GR↗
arXiv2019

Substitutes for the non-existent square lattice designs for 36 varieties

Square lattice designs are often used in trials of new varieties of various agricultural crops. However, there are no square lattice designs for 36 varieties in blocks of size six for four or more replicates. Here we use three different approaches to construct designs for up to eight replicates. All the designs perform well in terms of giving a low average variance of variety contrasts. Supplementary materials are available online.

stat.ME↗
arXiv2019

A Survey on Varieties Generated by Small Semigroups and a Companion Website

The aim of this paper is to provide an atlas of identity bases for varieties generated by small semigroups and groups. To help the working mathematician easily find information, we provide a companion website that runs in the background automated reasoning tools, finite model builders, and GAP, so that the user has an automatic \textit{intelligent} guide on the literature. This paper is mainly a survey of what is known about identity bases for semigroups or groups of small orders, and we also mend some gaps left unresolved by previous authors. For instance, we provide the first complete and justified list of identity bases for the varieties generated by a semigroup of order up to~$4$, and the website contains the list of varieties generated by a semigroup of order up to~$5$. The website also provides identity bases for several types of semigroups or groups, such as bands, commutative groups, and metabelian groups. On the inherently non-finitely based finite semigroups side, the website can decide if a given finite semigroup possesses this property or not. We provide some other functionalities such as a tool that outputs the multiplication table of a semigroup given by a $C$-presentation, where~$C$ is any class of algebras defined by a set of first order formulas. The companion website can be found here \url{http://sgv.pythonanywhere.com} Please send any comments/suggestions to \url{jj.araujo@fct.unl.pt}

math.GR↗
arXiv2019

Primitive Permutation Groups and Strongly Factorizable Transformation Semigroups

Let $Ω$ be a finite set and $T(Ω)$ be the full transformation monoid on $Ω$. The rank of a transformation $t\in T(Ω)$ is the natural number $|Ωt|$. Given $A\subseteq T(Ω)$, denote by $\langle A\rangle$ the semigroup generated by $A$. Let $k$ be a fixed natural number such that $2\le k\le |Ω|$. In the first part of this paper we (almost) classify the permutation groups $G$ on $Ω$ such that for all rank $k$ transformation $t\in T(Ω)$, every element in $S_t:=\langle G,t\rangle$ can be written as a product $eg$, where $e^2=e\in S_t$ and $g\in G$. In the second part we prove, among other results, that if $S\le T(Ω)$ and $G$ is the normalizer of $S$ in the symmetric group on $Ω$, then the semigroup $SG$ is regular if and only if $S$ is regular. (Recall that a semigroup $S$ is regular if for all $s\in S$ there exists $s'\in S$ such that $s=ss's$.) The paper ends with a list of problems.

math.GR↗
arXiv2019

Undirecting membership in models of ZFA

It is known that, if we take a countable model of Zermelo--Fraenkel set theory ZFC and "undirect" the membership relation (that is, make a graph by joining $x$ to $y$ if either $x\in y$ or $y\in x$), we obtain the Erdős--Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation, so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel's Anti-Foundation Axiom). The resulting graph may fail to be simple, it may have loops (if $x\in x$ for some $x$) or multiple edges (if $x\in y$ and $y\in x$ for some $x,y$). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the "random loopy graph" (which is $\aleph_0$-categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $\aleph_0$-categorical, but has infinitely many $1$-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.

math.LO↗
arXiv2019

The Hall--Paige conjecture, and synchronization for affine and diagonal groups

The Hall--Paige conjecture asserts that a finite group has a complete mapping if and only if its Sylow subgroups are not cyclic. The conjecture is now proved, and one aim of this paper is to document the final step in the proof (for the sporadic simple group $\mathrm{J}_4$). We apply this result to prove that primitive permutation groups of simple diagonal type with three or more simple factors in the socle are non-synchronizing. We also give the simpler proof that, for groups of affine type, or simple diagonal type with two socle factors, synchronization and separation are equivalent. Synchronization and separation are conditions on permutation groups which are stronger than primitivity but weaker than $2$-homogeneity, the second of these being stronger than the first. Empirically it has been found that groups which are synchronizing but not separating are rather rare. It follows from our results that such groups must be primitive of almost simple type.

math.GR↗
arXiv2018

The existential transversal property: a generalization of homogeneity and its impact on semigroups

Let $G$ be a permutation group of degree $n$, and $k$ a positive integer with $k\le n$. We say that $G$ has the $k$-existential property, or $k$-et for short, if there exists a $k$-subset $A$ of the domain $Ω$ such that, for any $k$-partition $\mathcal{P}$ of $Ω$, there exists $g\in G$ mapping $A$ to a transversal (a section) for $\mathcal{P}$. This property is a substantial weakening of the $k$-universal transversal property, or $k$-ut, investigated by the first and third author, which required this condition to hold for all $k$-subsets $A$ of the domain. Our first task in this paper is to investigate the $k$-et property and to decide which groups satisfy it. For example, we show that, for $8\le k\le n/2$, the only groups with $k$-et are the symmetric and alternating groups; this is best possible since the Mathieu group $M_{24}$ has $7$-et. We determine all groups with $k$-et for $4\le k\le n/2$, up to some unresolved cases for $k=4,5$, and describe the property for $k=2,3$ in permutation group language. In the previous work, the results were applied to semigroups, in particular, to the question of when the semigroup $\langle G,t\rangle$ is regular, where $t$ is a map of rank $k$ (with $k<n/2$); this turned out to be equivalent to the $k$-ut property. The question investigated here is when there is a $k$-subset $A$ of the domain such that $\langle G, t\rangle$ is regular for all maps $t$ with image $A$. This turns out to be more delicate; the $k$-et property (with $A$ as witnessing set) is a necessary condition, and the combination of $k$-et and $(k-1)$-ut is sufficient, but the truth lies somewhere between. Given the knowledge that a group under consideration has the necessary condition of $k$-et, we solve the regularity question for $k\le n/2$ except for one sporadic group.

math.GR↗
arXiv2018

Integrals of groups

An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are: (1) If a finite group has an integral, then it has a finite integral. (2) A precise characterization of the set of natural numbers $n$ for which every group of order $n$ is integrable: these are the cubefree numbers $n$ which do not have prime divisors $p$ and $q$ with $q\mid p-1$. (3) An abelian group of order $n$ has an integral of order at most $n^{1+o(1)}$, but may fail to have an integral of order bounded by $cn$ for constant $c$. (4) A finite group can be integrated $n$ times (in the class of finite groups) if and only if it is the central product of an abelian group and a perfect group. There are many other results on such topics as centreless groups, groups with composition length $2$, and infinite groups. We also include a number of open problems.

math.GR↗
arXiv2018

Multi-part balanced incomplete-block designs

We consider designs for cancer trials which allow each medical centre to treat only a limited number of cancer types with only a limited number of drugs. We specify desirable properties of these designs, and prove some consequences. Then we give several different constructions. Finally we generalise this to three or more types of object, such as biomarkers.

stat.AP↗
arXiv2018

Sesqui-arrays, a generalisation of triple arrays

A triple array is a rectangular array containing letters, each letter occurring equally often with no repeats in rows or columns, such that the number of letters common to two rows, two columns, or a row and a column are (possibly different) non-zero constants. Deleting the condition on the letters common to a row and a column gives a double array. We propose the term \emph{sesqui-array} for such an array when only the condition on pairs of columns is deleted. Thus all triple arrays are sesqui-arrays. In this paper we give three constructions for sesqui-arrays. The first gives $(n+1)\times n^2$ arrays on $n(n+1)$ letters for $n\geq 2$. (Such an array for $n=2$ was found by Bagchi.) This construction uses Latin squares. The second uses the \emph{Sylvester graph}, a subgraph of the Hoffman--Singleton graph, to build a good block design for $36$ treatments in $42$ blocks of size~$6$, and then uses this in a $7\times 36$ sesqui-array for $42$ letters. We also give a construction for $K\times(K-1)(K-2)/2$ sesqui-arrays on $K(K-1)/2$ letters. This construction uses biplanes. It starts with a block of a biplane and produces an array which satisfies the requirements for a sesqui-array except possibly that of having no repeated letters in a row or column. We show that this condition holds if and only if the \emph{Hussain chains} for the selected block contain no $4$-cycles. A sufficient condition for the construction to give a triple array is that each Hussain chain is a union of $3$-cycles; but this condition is not necessary, and we give a few further examples. We also discuss the question of which of these arrays provide good designs for experiments.

math.CO↗
arXiv2018

The power graph of a torsion-free group

The \emph{power graph} $P(G)$ of a group $G$ is the graph whose vertex set is $G$, with $x$ and $y$ joined if one is a power of the other; the \emph{directed power graph} $\vec{P}(G)$ has the same vertex set, with an arc from $x$ to $y$ if $y$ is a power of $x$. It is known that, for finite groups, the power graph determines the directed power graph up to isomorphism. However, it is not true that any isomorphism between power graphs induces an isomorphism between directed power graphs. Moreover, for infinite groups the power graph may fail to determine the directed power graph. In this paper, we consider power graphs of torsion-free groups. Our main results are that, for torsion-free nilpotent groups of class at most $2$, and for groups in which every non-identity element lies in a unique maximal cyclic subgroup, the power graph determines the directed power graph up to isomorphism. For specific groups such as $\mathbb{Z}$ and $\mathbb{Q}$, we obtain more precise results. Any isomorphism $P(\mathbb{Z})\to P(G)$ preserves orientation, so induces an isomorphism between directed power graphs; in the case of $\mathbb{Q}$, the orientations are either all preserved or all reversed. We also obtain results about groups in which every element is contained in a unique maximal cyclic subgroup (this class includes the free and free abelian groups), and about subgroups of the additive group of $\mathbb{Q}$ and about $\mathbb{Q}^n$.

math.GR↗
arXiv2017

Synchronization and separation in the Johnson schemes

Recently Peter Keevash solved asymptotically the existence question for Steiner systems by showing that $S(t,k,n)$ exists whenever the necessary divisibility conditions on the parameters are satisfied and $n$ is sufficiently large in terms of $k$ and $t$. The purpose of this paper is to make a conjecture which if true would be a significant extension of Keevash's theorem, and to give some theoretical and computational evidence for the conjecture. We phrase the conjecture in terms of the notions (which we define here) of synchronization and separation for association schemes. These definitions are based on those for permutation groups which grow out of the theory of synchronization in finite automata. In this theory, two classes of permutation groups (called \emph{synchronizing} and \emph{separating}) lying between primitive and $2$-homogeneous are defined. A big open question is how the permutation group induced by $S_n$ on $k$-subsets of $\{1,\ldots,n\}$ fits in this hierarchy; our conjecture would give a solution to this problem for $n$ large in terms of $k$. We prove the conjecture in the case $k=4$: our result asserts that $S_n$ acting on $4$-sets is separating for $n\ge10$ (it fails to be synchronizing for $n=9$).

math.GR↗
arXiv2017

On the automorphism group of the m-coloured random graph

Let $R_m$ be the (unique) universal homogeneous $m$-edge-coloured countable complete graph ($m\ge2$), and $G_m$ its group of colour-preserving automorphisms. The group $G_m$ was shown to be simple by John Truss. We examine the automorphism group of $G_m$, and show that it is the group of permutations of $R_m$ which induce permutations on the colours, and hence an extension of $G_m$ by the symmetric group of degree $m$. We show further that the extension splits if and only if $m$ is odd, and in the case where $m$ is even and not divisible by~$8$ we find the smallest supplement for $G_m$ in its automorphism group. (This unpublished paper from 2007 is placed here because of renewed interest in the topic.)

math.CO↗