Orbital graphs of primitive permutation groups

该项目旨在研究与本原置换群相关的某些图族，这些图是所有置换群的基本构件。如果G是作用于集合X上的传递作用的群，则X在笛卡儿积X×X上的每个轨道可视为具有顶点集X的图的边集，G作用于其上的边传递。这些称为与G相关的轨道图。D·希格曼的一个著名判据指出，所有的轨道图都是连通的当且仅当G在X上的作用是本原的。最近，人们对有限本原群的无限类进行了分类，对于这些本原群，所有轨道图的直径都有一个固定的常数。这项工作最初是由模型理论领域的问题推动的，模型理论是数理逻辑的一个分支-其中一个原因是，这样一个有界类中的群的超积是一个定义明确的对象，具有许多有趣的性质。刚才提到的分类在某种程度上是定性的--例如，给定一个显界d，它不会给出轨道直径有界于d的显式群族。这样的显式结果将是非常有趣的，并将产生新的密切相关的小直径边传递图族。即使对于d=2或3，这样的族也很少为人所知。该项目旨在明确地查找和分类这类族，并调查它们的一些基本属性。它是群论和代数组合学交界处的一个新方向，产生于数理逻辑中的应用。计划中的方法论将在很大程度上是代数的，是群论(主要是置换群和有限和代数单群)与代数图论的混合。在数学科学的战略主题中，它与EPSRC代数研究领域保持一致。

The project aims to study certain families of graphs associated with primitive permutation groups, which are the basic building blocks for all groups of permutations. If G is a group acting transitively on a set X, then each orbit of X on the Cartesian product X x X can be regarded as the set of edges of a graph with vertex set X, on which G acts edge-transitively. These are called the orbital graphs associated with G. A well-known criterion of D Higman states that the orbital graphs are all connected if and only if the action of G on X is primitive. There has been recent work on classifying infinite classes of finite primitive groups for which the diameters of all the orbital graphs are bounded by some fixed constant. This work was originally motivated by questions in the area of model theory, a branch of mathematical logic - one reason being that the ultraproduct of the groups in such a bounded class is a well-defined object with many interesting properties. The classification just mentioned is somewhat qualitative - for example, given an explicit bound d, it does not give explicit families of groups for which the orbital diameters are bounded by d. Such explicit results would be extremely interesting, and would give rise to new families of closely related edge-transitive graphs of small diameters. Even for d = 2 or 3, very few such families are known. This project aims to find and classify such families explicitly, and investigate some of their basic properties. It is a new direction at the interface between group theory and algebraic combinatorics, arising from applications in mathematical logic. The planned methodology will be largely algebraic, a mixture of group theory (mainly permutation groups and finite and algebraic simple groups), together with algebraic graph theory. It is aligned to the EPSRC Research Area of Algebra within the Strategic Theme of the Mathematical Sciences.