Invariable generation in finite groups with applications to algorithmic number theory

有限群中的不变生成及其在算法数论中的应用

基本信息

批准号：
EP/T017619/1
负责人：
Gareth Tracey
金额：
$ 36.77万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT017619%2F1
关键词：
Invariable generation finite groups applications

项目摘要

This research proposal lies at the interface of two areas of pure mathematics: algebra and number theory. More specifically, the research seeks to build on a fascinating link which has emerged between a problem in the theory of "groups" (the algebraic structures which capture and allow us to study symmetries in nature) and one of the most famous unsolved problems in mathematics: the Inverse Galois Problem.Galois theory was discovered by the French mathematician Evariste Galois in the nineteenth century as a tool to study (integer) polynomial equations of degree greater than 4, and when they can be solved by radicals. To a set of roots of such a polynomial, Galois associated an algebraic structure which we now call a Galois group. This structure is a set together with a binary operation which preserves the symmetries in the roots of the polynomial, and studying this operation allows us to deduce properties of the roots. In this way, Galois developed a theory whereby one can translate questions about a very complicated polynomial equation to questions about its Galois group, which is often easier and more concise to study. In recent years, the theory, together with group theory and number theory in general, has shifted from being a purely academic endeavour to making significant contributions to cryptography, e-commerce and financial security. Galois groups are special examples of the "groups" we mentioned in the first paragraph, and have finite size. Thus, all Galois groups are finite groups, but what about the other way around? Is every finite group the Galois group of some integer polynomial? This is called the "Inverse Galois Problem" (IGP), and a complete solution has evaded mathematicians for almost 200 years. A groups which is the Galois group of an integer polynomial is said to "satisfy IGP". Some specific cases have been dealt with (the group of all symmetries of a finite set of size n - the symmetric group of degree n - is known to satisfy the IGP, for example), but even some relatively "small" groups remain elusive. In 2008, the number theorists Jouve, Kowlaski and Zywina announced a new technique to study the IGP in the Weyl groups of simple algebraic groups - an important class of groups in geometry. This spawned a renewed optimism for the IGP, and led to further developments of the technique by Lucchini and Tracey (using powerful group theoretic techniques) and the mathematicians Eberhard, Ford and Green (using powerful techniques from probability theory and combinatorics). This research proposal seeks to build on these techniques by combining the group theoretic, probabilistic, and combinatorial approaches mentioned above. The specific problems we propose range from answering important questions concerning these techniques in the finite simple groups (the "building blocks" of finite groups), to answering some long-standing questions posed by B.L. van der Waerden and J.P. Serre. As a final ambitious problem, we seek to solve the IGP in the case when G is the Mathieu group M23 - one of the most famous and important finite groups in which it is unknown whether or not the IGP is satisfied.

该研究建议在于纯数学的两个领域的界面：代数和数理论。更具体地说，该研究试图建立在“群体”理论（捕获并使我们可以在自然界研究对称性的代数结构）中出现的有趣的联系，这是数学中最著名的未解决问题之一：逆向加洛伊斯问题。Galois理论是由19世纪的法国数学家Evariste Galois发现的，作为研究（整数）多项式学位的工具，即大于4的多项式方程，并且可以通过自由基解决。与一组此类多项式的根，Galois关联了一个代数结构，我们现在称之为Galois组。该结构与二进制操作在一起，可保留多项式根部的对称性，研究此操作使我们能够推断根的性质。通过这种方式，加洛伊斯（Galois）开发了一种理论，可以将有关非常复杂的多项式方程式的问题转化为有关其Galois群体的问题，该问题通常更容易，更简洁。近年来，该理论以及一般的群体理论和数字理论从纯粹的学术努力转变为对加密，电子商务和财务安全做出重大贡献。 Galois组是我们第一段中提到的“组”的特殊示例，并且规模有限。因此，所有加洛伊斯群体都是有限的群体，但是相反的情况呢？每个有限的组是否有一些整数多项式的Galois组？这被称为“逆向加洛伊斯问题”（IGP），一个完整的解决方案已经逃避了将近200年的数学家。据说是整数多项式的Galois组的组“满足IGP”。已经处理了一些具体情况（例如，n- n- n- n-的对称组的所有对称组的组 - 例如，n-的对称群体可以满足IGP的满足），但即使是一些相对“小”的组仍然难以捉摸。 2008年，数字理论家Jouve，Kowlaski和Zywina宣布了一项新技术，以研究简单代数组的Weyl群的IGP，这是一类重要的几何组。这催生了对IGP的新乐观，并导致了Lucchini和Tracey（使用强大的群体理论技术）和数学家Eberhard，Ford和Green（使用概率理论和组合学的强大技术）对技术的进一步发展。这项研究建议试图通过结合上述群体理论，概率和组合方法来基于这些技术。我们提出的具体问题包括从有限的简单组中回答有关这些技术的重要问题（有限群体的“构件”）到回答B.L.提出的一些长期存在的问题。 van der Waerden和J.P. Serre。作为最后一个雄心勃勃的问题，我们试图在G是Mathieu集团M23的情况下解决IGP，这是最著名，最重要的有限群体之一，其中未知IGP是否满足。