Orbital graphs of primitive permutation groups

本原排列群的轨道图

• 批准号: 2280025
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2280025
• 关键词: Orbital; graphs; primitive; permutation; groups

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.

该项目旨在研究与原始置换群相关的某些图族，这些图族是所有置换群的基本构建块。如果G是作用于集合X的传递群，则X在笛卡尔积X X上的每一个轨道都可以看作是顶点集X的图的边的集合，G作用于这个图的边传递。这些被称为与G相关的轨道图，一个著名的D Higman准则指出，当且仅当G对X的作用是基元时，轨道图都是连通的。最近有关于无限类有限原始群的分类的工作，其中所有轨道图的直径都由某个固定常数限定。这项工作最初是由数学逻辑的一个分支模型理论领域的问题所激发的——一个原因是，在这样一个有界的类中，群的超积是一个定义良好的对象，具有许多有趣的性质。刚才提到的分类有点定性——例如，给定一个明确的界d，它并没有给出轨道直径以d为界的群的明确族。这种明确的结果将是非常有趣的，并且会产生新的紧密相关的小直径边传递图族。即使对于d = 2或3，也很少有这样的家族是已知的。本项目旨在明确地发现和分类这些家族，并研究它们的一些基本属性。它是群论与代数组合学相结合的一个新方向，是在数理逻辑中的应用而产生的。计划的方法将主要是代数的，是群论（主要是置换群、有限群和代数单群）与代数图论的混合。它与EPSRC数学科学战略主题中的代数研究领域保持一致。