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/T017619/3
负责人：
Gareth Tracey
金额：
$ 20.73万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT017619%2F3
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.

这项研究提案位于纯数学两个领域的交汇处：代数和数论。更具体地说，该研究旨在建立“群”理论（捕捉并允许我们研究自然界对称性的代数结构）问题与数学中最著名的未解决问题之一之间出现的令人着迷的联系。：逆伽罗瓦问题。伽罗瓦理论是由法国数学家 Evariste Galois 在 19 世纪发现的，作为研究次数大于 4 的（整数）多项式方程的工具，并且当它们可以用根式解决时。对于这样一个多项式的一组根，伽罗瓦关联了一个代数结构，我们现在称之为伽罗瓦群。该结构是与二元运算的集合，二元运算保留了多项式根的对称性，研究该运算使我们能够推断出根的性质。通过这种方式，伽罗瓦发展了一种理论，通过该理论，人们可以将有关非常复杂的多项式方程的问题转化为有关其伽罗瓦群的问题，这通常更容易、更简洁地研究。近年来，该理论与群论和一般数论一起，已经从纯粹的学术努力转变为对密码学、电子商务和金融安全做出了重大贡献。伽罗瓦群是我们在第一段中提到的“群”的特殊例子，并且具有有限的大小。因此，所有伽罗瓦群都是有限群，但反过来呢？每个有限群都是某个整数多项式的伽罗瓦群吗？这被称为“逆伽罗瓦问题”（IGP），数学家们近 200 年来一直没有找到完整的解决方案。作为整数多项式的伽罗瓦群的群被称为“满足IGP”。一些特定的情况已经得到处理（例如，大小为 n 的有限集的所有对称性的群 - 度为 n 的对称群 - 已知满足 IGP），但即使是一些相对“小的”群仍然难以捉摸。 2008 年，数论学家 Jouve、Kowlaski 和 Zywina 宣布了一项新技术，用于研究简单代数群（几何中一类重要的群）的 Weyl 群中的 IGP。这给 IGP 带来了新的乐观情绪，并导致 Lucchini 和 Tracey（使用强大的群论技术）以及数学家 Eberhard、Ford 和 Green（使用概率论和组合学的强大技术）进一步发展该技术。本研究提案旨在通过结合上述群论、概率和组合方法来建立这些技术。我们提出的具体问题范围从回答有关有限单群（有限群的“构建块”）中这些技术的重要问题，到回答 B.L. 提出的一些长期存在的问题。范德瓦尔登和 J.P.塞尔。作为最后一个雄心勃勃的问题，我们寻求在 G 是马蒂厄群 M23 的情况下解决 IGP——最著名和最重要的有限群之一，其中 IGP 是否满足是未知的。