The book is aimed at postgraduate students and researchers into finite groups, although most of the material covered will be comprehensible to fourth year undergraduates who have taken two modules of group theory. It is based on the author's technique of symmetric generation, which seems able to present many difficult group-theoretic constructions in a more elementary manner. It is thus the aim of the book to make these beautiful, but combinatorially complicated, objects accessible to a wider audience.

The stimulus for the investigations which led to the contents of the book was a question from a colleague of mine, Tony Gardiner, who asked me if the Mathieu group M24 could contain two copies of the linear group L3(2) which intersect in a subgroup isomorphic to the symmetric group S4. He needed such a configuration in order to construct a graph with certain desirable properties. I assured him that the answer was almost certainly yes, but that I would work out the details. I decided to use copies of L3(2) which are maximal in M24 and found that the required intersection occurred in the nicest possible way, in that one could find subgroups H ≅ K ≅ L3(2), with H ∩ K ≅ S4, and an involution t such that CM24(H ∩ K) = 〈t〈 and Ht = K. This means that t has seven images under conjugation by H, and the maximality of H together with the simplicity of M24 mean that these seven involutions must generate M24. The symmetry of the whole set-up enables one to write down seven corresponding involutory permutations on 24 letters directly from a consideration of the action of L3(2) on 24 points.

Abstract

How many points are there on a curve with coordinates in a given finite field when the curve has (a) no singular points or (b) singular points counted once or (c) singular points counted with multiplicity?

What is the maximum number of points on a curve of given genus?

Can curves attaining this maximum number be characterised?

Introduction

Problems in combinatorics, especially in finite geometry, often require a count of the number of solutions of an equation in one or more unknowns defined over a finite field Fq. When two unknowns, say X, Y, occur, the equation is of type f(X, Y) = 0 with f ∈ Fq[X, Y], and the geometric approach for solving it depends on the theory of algebraic curves over finite fields.

Curves over a finite field have applications in the theory of linear error-correcting codes in two areas: (a) the construction of Goppa or algebraic-geometry codes; (b) obtaining bounds for the maximum length of codes when given the dimension and minimum distance.

In cryptography, ciphers are constructed from both elliptic and hyperelliptic curves

It is natural to think about a plane algebraic curve F of equation f(X, Y) = 0 as the set of the points P = (x, y) in the affine plane over the coordinate field K such that f(x, y) = 0. But important numerical results on curves and their intersections, such as Bézout's theorem, have an easier formulation when the following are taken into consideration.

The Twenty-first Biennial British Combinatorial Conference was held at Reading in July 2007. The British Combinatorial Committee had invited ten distinguished combinatorial mathematicians to give survey lectures in areas of their expertise, and this volume contains the survey articles on which these lectures were based.

In compiling this volume we are indebted to the authors for preparing their articles so accurately and in such a timely manner and to the referees for their attention to detail while commenting on the articles. We would also like to thank Roger Astley at Cambridge University Press for his advice and help.

The British Combinatorial Committee gratefully acknowledges the financial support provided by the London Mathematical Society, the Institue for Combinatorics and its Applications, and the EPSRC.

Introduction.

The notion of branchwidth was introduced by Robertson and Seymour in their seminal paper Graph Minors X [3]. Very roughly speaking, the goal is to decompose a structure S along a tree T in such a way that subsets of S corresponding to disjoint branches of T are pairwise as disjoint as possible. One can define the branchwidth of various structures such as graphs, hypergraphs, matroids, submodular functions … Our goal in this paper is to prove that the definitions of branchwidth for graphs and matroids coincide in the sense that the branchwidth of a bridgeless graph is equal to the branchwidth of its cycle matroid. This answers a question of Thomas [5], also cited in Geelen, Gerards, Robertson and Whittle [1].

Let us now define properly these notions.

Let H = (V, E) be a graph, or a hypergraph, and (E1, E2) be a partition of E. The border of (E1, E2) is the set of vertices which belong to both an edge of E1 and an edge of E2. We denote this by δ(E1, E2), or simply by δ(E1).

A branch-decomposition T of H is a ternary tree T and a bijection from the set of leaves of T into the set of edges of H.

Abstract

A hereditary property of graphs is a collection of (isomorphism classes of) graphs which is closed under taking induced graphs, and contains arbitrarily large structures. Given a family F of graphs, the family P(F) of graphs containing no member of F as an induced subgraph is a hereditary property, and every hereditary property of graphs arises in this way. A hereditary property of other combinatorial structures is defined analogously. A property is monotone if it is closed under taking (not necessarily induced) substructures.

Given a property P, we write Pn for the number of distinct structures with vertices labelled 1, …, n, and call the function n ↦ |Pn| the labelled speed of P. Similarly, the unlabelled speed is n ↦ |Pn|, where Pn is the set of distinct structures with n unlabelled vertices. The study of hereditary properties is on the borderline of extremal, enumerative, and probabilistic combinatorics. Thus, for a family F of graphs, the problem of determining the speed of P(F) is a natural extension of the basic question in extremal graph theory concerned with the maximal number of edges in a graph of order n containing no member of F as a subgraph.

For many a combinatorial structure (graphs, posets, partitions, words, etc.), there is a surprising phase transition: the speed jumps from one range to a much higher one. Thus the speed of a property is either not much larger than a certain function f(n) or is at least as large as a function F which is much larger than f.

Introduction

Cryptography provides the core information security services that are necessary to safeguard electronic communications. The sound management of cryptographic keys is the fundamental supporting activity that underpins the secure implementation of cryptography. The purpose of this paper is to demonstrate the significant contribution of combinatorial mathematics to the development of the theory of cryptographic key establishment.

• Scope: This paper surveys areas of key establishment where combinatorial models or construction techniques have proven of value. Our aim is not to provide a comprehensive survey of the literature, but rather to provide sufficient coverage that most relevant work will be (to use the terminology of Section 7.3.3) at most a “two-hop path” from this review. This paper is not an attempt to survey the vast research on key establishment in general.

• Detail: The primary aim is to bring these applications of combinatorics to the attention of the mathematical community within a sensible unifying framework. We thus focus on introducing concepts and providing pointers for further study. This paper contains no proofs. Combinatorial modelling typically involves the establishment of bounds and constructions. For illustrative purposes we will tend to focus on constructions in this review.

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