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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/T017619/2
负责人：
Gareth Tracey
金额：
$ 26.18万
依托单位：
University of Birmingham
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT017619%2F2
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.

这项研究计划是在纯数学的两个领域的接口：代数和数论。更具体地说，这项研究试图建立在一个迷人的联系，这已经出现了一个问题之间的理论“群体”（代数结构，捕获并允许我们研究自然界中的对称性）和数学中最著名的未解决的问题之一：伽罗瓦逆问题。伽罗瓦理论是由法国数学家埃瓦里斯特·伽罗瓦在十九世纪发现的，作为研究（整数）的工具。多项式方程的次数大于4，当他们可以解决的自由基。对于这样一个多项式的一组根，伽罗瓦给出了一个代数结构，我们现在称之为伽罗瓦群。这个结构是一个集合，它与一个二元运算一起保持多项式根的对称性，研究这个运算可以让我们推导出根的性质。通过这种方式，伽罗瓦发展了一种理论，人们可以将关于一个非常复杂的多项式方程的问题转化为关于其伽罗瓦群的问题，这通常更容易和更简洁地研究。近年来，该理论与群论和数论一起，已经从纯粹的学术努力转向对密码学，电子商务和金融安全做出重大贡献。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，所有的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？这就是所谓的“伽罗瓦逆问题”（IGP），一个完整的解决方案已经逃避了近200年的数学家。一个整数多项式的伽罗瓦群称为“满足IGP”。一些特殊的情况已经被处理过（例如，一个大小为n的有限集合的所有对称的群--n次对称群--已知满足IGP），但即使是一些相对“小”的群仍然是难以捉摸的。2008年，数论学家Jouve、Kowlaski和Zywina宣布了一种新的技术来研究简单代数群的Weyl群中的IGP，简单代数群是几何中的一类重要群。这使人们对IGP重新感到乐观，并导致卢基尼和特雷西（使用强大的群论技术）以及数学家埃伯哈德、福特和绿色（使用概率论和组合学的强大技术）对该技术的进一步发展。本研究建议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的范围从回答有关这些技术在有限简单群（有限群的“积木”）的重要问题，回答一些长期存在的问题提出的B.L.货车·德尔·瓦尔登和J·P·塞尔。作为最后一个雄心勃勃的问题，我们寻求解决的情况下，当G是马蒂厄群M23 -最著名的和重要的有限群之一，它是未知的IGP是否满足。