On Cherlin's conjecture for finite binary primitive permutation groups

关于有限二元本原置换群的 Cherlin 猜想

• 批准号: EP/R028702/1
• 负责人: Nick Gill
• 金额: $ 12.41万
• 依托单位: University of South Wales
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR028702%2F1
• 关键词: Cherlin; conjecture; finite; binary; primitive

This research concerns the connection between the "local symmetry" and "global symmetry" of a mathematical object. To understand what this means let us consider the symmetries of a simple mathematical object: a regular hexagon. If I randomly pick two different sides of this hexagon, then it is clear that they "look the same" -- this just follows from the fact that the hexagon is regular and so all sides have the same length. This is an example of a "local symmetry" -- that's the mathematical terminology for a situation where two portions of a mathematical object that look the same. On the other hand, because the hexagon is regular there are many "global symmetries" -- these are transformations of my object so that after the transformation it still "looks the same". In the case of the regular hexagon, for instance, I can reflect the hexagon through a line connecting two opposite corners, and the resulting object will be the same as the one I started with. Likewise, I can rotate my hexagon by any multiple of 60 degrees and the same will be true -- these are all examples of global symmetries.Now a natural question that mathematicians want to know when they study any given mathematical object is "when does the presence of a local symmetry imply the presence of a global symmetry?". For instance in the example above, it is clear that given any pair of edges on my hexagon, I can find a global symmetry (a rotation, for example) that moves the first edge to the second edge. Thus we could say that the local symmetry here is just a consequence of the global symmetry. In fact this is rather unusual: most mathematical objects will have many local symmetries that are NOT consequences of some global symmetry. A mathematical object for which all local symmetries derive from global symmetries is very special, and is called HOMOGENEOUS.The research in this project concerns homogeneous RELATIONAL STRUCTURES. A relational structure is just a particular generalization of a network: take a bunch of "nodes" and connect them with "edges" and you have made yourself a network (think of cities connected by roads, or computers connected by wires for real-life examples). Indeed the hexagon can be thought of as a network -- each corner can be thought of as a node (there are 6 of these), and then there are 6 edges connecting the nodes. We would like to know which networks are homogeneous. To fully understand this question, one needs to be a bit careful about how we define the notion of "symmetry" for a network and there is not time to do this here. As a teaser, though, let us mention that the network given by a hexagon is NOT homogeneous, whereas the network given by a pentagon IS!Finally, let us say a word about our methods: whenever one studies symmetry in mathematics, one is effectively doing GROUP THEORY. The set of symmetries of any mathematical object (say the reflections and rotations of our regular hexagon), is called the GROUP associated to the object. One can study this group "in the abstract", i.e. without really needing to study the object it is associated with. For instance, if I perform a particular reflection and then a particular rotation of my hexagon, I will end up with a new symmetry of the hexagon (in fact it will be another of the reflections) and I can think of this as a type of "multiplication" of my symmetries: I've "multiplied" two symmetries together and the result is a third symmetry. To fully describe the symmetry group of my hexagon I just need to write down the "multiplication table" of all pairs of symmetries.A great deal is known about the structure of different groups. Indeed one of the most famous and important mathematical theorems is called THE CLASSIFICATION OF FINITE SIMPLE GROUPS and it describes the structure of an important class of groups. In this research we will use this theorem to study homogeneous relational structures; our aim is to classify an important subclass of these objects.

本研究关注数学对象的“局部对称性”和“整体对称性”之间的联系。为了理解这意味着什么，让我们考虑一个简单的数学对象的对称性：正六边形。如果我随机选择这个六边形的两个不同的边，那么很明显它们“看起来一样”--这是因为六边形是规则的，所以所有的边都有相同的长度。这是一个“局部对称”的例子--这是一个数学术语，指的是一个数学对象的两个部分看起来相同的情况。另一方面，因为六边形是规则的，所以有许多“整体对称性”--这些是我的对象的变换，所以在变换之后它仍然“看起来一样”。例如，在正六边形的情况下，我可以通过连接两个对角的直线反射六边形，结果对象将与我开始时的对象相同。同样地，我可以将六边形旋转60度的任意倍数，这也是正确的--这些都是整体对称的例子。现在，数学家们在研究任何给定的数学对象时，都想知道的一个自然问题是“什么时候局部对称的存在意味着整体对称的存在？".例如，在上面的例子中，很明显，给定我的六边形上的任何一对边，我可以找到一个全局对称（例如旋转），将第一条边移动到第二条边。因此，我们可以说，这里的局部对称性只是全局对称性的一个结果。事实上，这是相当不寻常的：大多数数学对象将有许多局部对称性，而这些局部对称性不是某些整体对称性的结果。一个数学对象的所有局部对称性都来自于整体对称性，这是一个非常特殊的数学对象，我们称之为齐次关系结构。关系结构只是网络的一种特殊概括：取一堆“节点”，用“边”将它们连接起来，你就构成了一个网络（想想用道路连接的城市，或者用电线连接的计算机）。事实上，六边形可以被认为是一个网络-每个角可以被认为是一个节点（有6个），然后有6条边连接节点。我们想知道哪些网络是同质的。为了充分理解这个问题，我们需要对如何定义网络的“对称性”概念有点谨慎，这里没有时间这样做。不过，作为一个预告片，让我们提到六边形给出的网络不是均匀的，而五边形给出的网络是均匀的！最后，让我们对我们的方法说一句话：每当一个人研究数学中的对称性时，他实际上是在做群论。任何数学对象的对称性集合（比如正六边形的反射和旋转）称为与该对象相关的群。人们可以“抽象地”研究这一群体，即不需要真正研究与之相关的对象。例如，如果我执行一个特定的反射，然后对我的六边形进行一个特定的旋转，我最终会得到一个新的六边形对称（实际上它将是另一个反射），我可以把这看作是我的对称的一种“乘法”：我把两个对称“相乘”在一起，结果是第三个对称。为了完整地描述我的六边形的对称群，我只需要写下所有对称对的“乘法表”。事实上，最著名的和重要的数学定理之一是所谓的分类有限简单群，它描述了结构的一个重要类别的群体。在这项研究中，我们将使用这个定理来研究同构关系结构，我们的目标是对这些对象的一个重要子类进行分类。

项目成果

Cherlin's Conjecture for Finite Primitive Binary Permutation Groups

切尔林有限原二元置换群猜想

• DOI: 10.1007/978-3-030-95956-2
• 发表时间: 2022
• 作者: Gill N

Binary permutation groups: Alternating and classical groups

二元排列群：交替群和经典群

• DOI: 10.1353/ajm.2020.0000
• 发表时间: 2020
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: Gill N

Cherlin's conjecture for almost simple groups of Lie rank 1

Cherlin 对李阶 1 的几乎简单群的猜想

• DOI: 10.1017/s0305004118000403
• 发表时间: 2018
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: GILL N

Statistics for S acting on k-sets

S 作用于 k 集的统计

• DOI: 10.1016/j.jalgebra.2021.10.037
• 发表时间: 2022
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Gill N

ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

论有限排列群的高度和关系复杂性

• DOI: 10.1017/nmj.2021.6
• 发表时间: 2021
• 期刊: Nagoya Mathematical Journal
• 影响因子: 0.8
• 作者: GILL N