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Peter J. Cameron

Publications and source records attributed to Peter J. Cameron.

At least 109 records · Page 6Linked to original sources
arXiv2014

Most primitive groups are full automorphism groups of edge-transitive hypergraphs

We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,Y^G) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n^{1/2+ε} for the minimum size of the edges in such a hypergraph. This is essentially best possible.

math.GR↗
arXiv2014

A graph partition problem

Given a graph $G$ on $n$ vertices, for which $m$ is it possible to partition the edge set of the $m$-fold complete graph $mK_n$ into copies of $G$? We show that there is an integer $m_0$, which we call the \emph{partition modulus of $G$}, such that the set $M(G)$ of values of $m$ for which such a partition exists consists of all but finitely many multiples of $m_0$. Trivial divisibility conditions derived from $G$ give an integer $m_1$ which divides $m_0$; we call the quotient $m_0/m_1$ the \emph{partition index of $G$}. It seems that most graphs $G$ have partition index equal to $1$, but we give two infinite families of graphs for which this is not true. We also compute $M(G)$ for various graphs, and outline some connections between our problem and the existence of designs of various types.

math.CO↗
arXiv2014

Two Generalizations of Homogeneity in Groups with Applications to Regular Semigroups

Let $X$ be a finite set such that $|X|=n$ and let $i\leq j \leq n$. A group $G\leq \sym$ is said to be $(i,j)$-homogeneous if for every $I,J\subseteq X$, such that $|I|=i$ and $|J|=j$, there exists $g\in G$ such that $Ig\subseteq J$. (Clearly $(i,i)$-homogeneity is $i$-homogeneity in the usual sense.) A group $G\leq \sym$ is said to have the $k$-universal transversal property if given any set $I\subseteq X$ (with $|I|=k$) and any partition $P$ of $X$ into $k$ blocks, there exists $g\in G$ such that $Ig$ is a section for $P$. (That is, the orbit of each $k$-subset of $X$ contains a section for each $k$-partition of $X$.) In this paper we classify the groups with the $k$-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the $(k-1,k)$-homogeneous groups (for $2<k\leq \lfloor \frac{n+1}{2}\rfloor$). As a corollary of the classification we prove that a $(k-1,k)$-homogeneous group is also $(k-2,k-1)$-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the $k$-universal transversal property have the $(k-1)$-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank $k$ transformation on $X$ generate a regular semigroup (for $1\leq k\leq \lfloor \frac{n+1}{2}\rfloor$). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.

math.GR↗
arXiv2014

Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks

Let $Ω$ be a set of cardinality $n$, $G$ a permutation group on $Ω$, and $f:Ω\toΩ$ a map which is not a permutation. We say that $G$ synchronizes $f$ if the semigroup $\langle G,f\rangle$ contains a constant map. The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree $n$ primitive groups synchronize maps of rank $n-1$ (thus, maps with kernel type $(2,1,\ldots,1)$). We prove some extensions of Rystsov's result, including this: a primitive group synchronizes every map whose kernel type is $(k,1,\ldots,1)$. Incidentally this result provides a new characterization of imprimitive groups. We also prove that the conjecture above holds for maps of extreme ranks, that is, ranks 3, 4 and $n-2$. These proofs use a graph-theoretic technique due to the second author: a transformation semigroup fails to contain a constant map if and only if it is contained in the endomorphism semigroup of a non-null (simple undircted) graph. The paper finishes with a number of open problems, whose solutions will certainly require very delicate graph theoretical considerations.

math.CO↗
arXiv2013

Computing in permutation groups without memory

Memoryless computation is a new technique to compute any function of a set of registers by updating one register at a time while using no memory. Its aim is to emulate how computations are performed in modern cores, since they typically involve updates of single registers. The memoryless computation model can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we consider how efficiently permutations can be computed without memory. We determine the minimum number of basic updates required to compute any permutation, or any even permutation. The small number of required instructions shows that very small instruction sets could be encoded on cores to perform memoryless computation. We then start looking at a possible compromise between the size of the instruction set and the length of the resulting programs. We consider updates only involving a limited number of registers. In particular, we show that binary instructions are not enough to compute all permutations without memory when the alphabet size is even. These results, though expressed as properties of special generating sets of the symmetric or alternating groups, provide guidelines on the implementation of memoryless computation.

cs.CC↗
arXiv2013

Computing in matrix groups without memory

Memoryless computation is a novel means of computing any function of a set of registers by updating one register at a time while using no memory. We aim to emulate how computations are performed on modern cores, since they typically involve updates of single registers. The computation model of memoryless computation can be fully expressed in terms of transformation semigroups, or in the case of bijective functions, permutation groups. In this paper, we view registers as elements of a finite field and we compute linear permutations without memory. We first determine the maximum complexity of a linear function when only linear instructions are allowed. We also determine which linear functions are hardest to compute when the field in question is the binary field and the number of registers is even. Secondly, we investigate some matrix groups, thus showing that the special linear group is internally computable but not fast. Thirdly, we determine the smallest set of instructions required to generate the special and general linear groups. These results are important for memoryless computation, for they show that linear functions can be computed very fast or that very few instructions are needed to compute any linear function. They thus indicate new advantages of using memoryless computation.

cs.CC↗
arXiv2013

Permutation groups and transformation semigroups: results and problems

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group $G$ and a single non-permutation $a$. Our particular concern is the influence that properties of $G$ (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of $S= $ such as regularity and idempotent generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances $S$ contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided. These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.

math.GR↗
arXiv2013

The Classification of Partition Homogeneous Groups with Applications to Semigroup Theory

Let $λ=(λ_1,λ_2,...)$ be a \emph{partition} of $n$, a sequence of positive integers in non-increasing order with sum $n$. Let $Ω:=\{1,...,n\}$. An ordered partition $P=(A_1,A_2,...)$ of $Ω$ has \emph{type} $λ$ if $|A_i|=λ_i$. Following Martin and Sagan, we say that $G$ is \emph{$λ$-transitive} if, for any two ordered partitions $P=(A_1,A_2,...)$ and $Q=(B_1,B_2,...)$ of $Ω$ of type $λ$, there exists $g\in G$ with $A_ig=B_i$ for all $i$. A group $G$ is said to be \emph{$λ$-homogeneous} if, given two ordered partitions $P$ and $Q$ as above, inducing the sets $P'=\{A_1,A_2,...\}$ and $Q'=\{B_1,B_2,...\}$, there exists $g\in G$ such that $P'g=Q'$. Clearly a $λ$-transitive group is $λ$-homogeneous. The first goal of this paper is to classify the $λ$-homogeneous groups. The second goal is to apply this classification to a problem in semigroup theory. Let $\trans$ and $\sym$ denote the transformation monoid and the symmetric group on $Ω$, respectively. Fix a group $H\leq \sym$. Given a non-invertible transformation $a\in \trans\setminus \sym$ and a group $G\leq \sym$, we say that $(a,G)$ is an \emph{$H$-pair} if the semigroups generated by $\{a\}\cup H$ and $\{a\}\cup G$ contain the same non-units, that is, $< a,G>\setminus G=< a,H>\setminus H$. Using the classification of the $λ$-homogeneous groups we classify all the $\sym$-pairs. This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas. The paper finishes with a number of open problems on permutation and linear groups.

math.GR↗
arXiv2013

The random graph

Erdős and Rényi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.

math.CO↗
arXiv2012

The classification of normalizing groups

Let $X$ be a finite set such that $|X|=n$. Let $\trans$ and $\sym$ denote respectively the transformation monoid and the symmetric group on $n$ points. Given $a\in \trans\setminus \sym$, we say that a group $G\leq \sym$ is $a$-normalizing if $$ \setminus G= .$$ If $G$ is $a$-normalizing for all $a\in \trans\setminus \sym$, then we say that $G$ is normalizing. The goal of this paper is to classify normalizing groups and hence answer a question posed elsewhere. The paper ends with a number of problems for experts in groups, semigroups and matrix theory.

math.GR↗
arXiv2012

Groups with right-invariant multiorders

A Cayley object for a group G is a structure on which G acts regularly as a group of automorphisms. The main theorem asserts that a necessary and sufficient condition for the free abelian group G of rank m to have the generic n-tuple of linear orders as a Cayley object is that m>n. The background to this theorem is discussed. The proof uses Kronecker's Theorem on diophantine approximation.

math.CO↗
arXiv2012

Groups Synchronizing a Transformation of Non-Uniform Kernel

This paper concerns the general problem of classifying the finite deterministic automata that admit a synchronizing (or reset) word. (For our purposes it is irrelevant if the automata has initial or final states.) Our departure point is the study of the transition semigroup associated to the automaton, taking advantage of the enormous and very deep progresses made during the last decades on the theory of permutation groups, their geometry and their combinatorial structure. Let $X$ be a finite set. We say that a primitive group $G$ on $X$ is {\em synchronizing} if $G$ together with any non-invertible map on $X$ generates a constant map. It is known (by some recent results proved by P. M. Neumann) that for some primitive groups $G$ and for some singular transformations $t$ of uniform kernel (that is, all blocks have the same number of elements), the semigroup $< G,t>$ does not generate a constant map. Therefore the following concept is very natural: a primitive group $G$ on $X$ is said to be {\em almost synchronizing} if $G$ together with any map of non-uniform kernel generates a constant map. In this paper we use two different methods to provide several infinite families of groups that are not synchronizing, but are almost synchronizing. The paper ends with a number of problems on synchronization likely to attract the attention of experts in computer science, combinatorics and geometry, groups and semigroups, linear algebra and matrix theory.

math.GR↗
arXiv2012

A Unifying Framework to Characterize the Power of a Language to Express Relations

In this extended abstract we provide a unifying framework that can be used to characterize and compare the expressive power of query languages for different data base models. The framework is based upon the new idea of valid partition, that is a partition of the elements of a given data base, where each class of the partition is composed by elements that cannot be separated (distinguished) according to some level of information contained in the data base. We describe two applications of this new framework, first by deriving a new syntactic characterization of the expressive power of relational algebra which is equivalent to the one given by Paredaens, and subsequently by studying the expressive power of a simple graph-based data model.

cs.DB↗
arXiv2012

Remoteness of permutation codes

In this paper, we introduce a new parameter of a code, referred to as the remoteness, which can be viewed as a dual to the covering radius. Indeed, the remoteness is the minimum radius needed for a single ball to cover all codewords. After giving some general results about the remoteness, we then focus on the remoteness of permutation codes. We first derive upper and lower bounds on the minimum cardinality of a code with a given remoteness. We then study the remoteness of permutation groups. We show that the remoteness of transitive groups can only take two values, and we determine the remoteness of transitive groups of odd order. We finally show that the problem of determining the remoteness of a given transitive group is equivalent to determining the stability number of a related graph.

math.CO↗
arXiv2011

Aftermath

I take a quick overview at the recent development of combinatorics and its current directions, as a discipline in its own right, as part of mathematics, and as part of science and wider society.

math.HO↗
arXiv2011

Using graphs to find the best block designs

A statistician designing an experiment wants to get as much information as possible from the data gathered. Often this means the most precise estimate possible (that is, an estimate with minimum possible variance) of the unknown parameters. If there are several parameters, this can be interpreted in many ways: do we want to minimize the average variance, or the maximum variance, or the volume of a confidence region for the parameters? In the case of block designs, these optimality criteria can be calculated from the concurrence graph of the design, and in many cases from its Laplacian eigenvalues. The Levi graph can also be used. The various criteria turn out to be closely connected with other properties of the graph as a network, such as number of spanning trees, isoperimetric number, and the sum of the resistances between pairs of vertices when the graph is regarded as an electrical network. In this chapter, we discuss the notions of optimality for incomplete-block designs, explain the graph-theoretic connections, and prove some old and new results about optimality.

math.ST↗
arXiv2011

Galois groups of multivariate Tutte polynomials

The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let $\bv = \{v_e\}_{e\in E}$, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that $\hat Z_M$ is irreducible over $K(\bv)$, and give three self-contained proofs that the Galois group of $\hat Z_M$ over $K(\bv)$ is the symmetric group of degree $n$, where $n$ is the rank of $M$. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.

math.CO↗