Invariable generation in finite groups with applications to algorithmic number theory

本研究计划涉及纯数学的两个领域：代数和数论。更具体地说，这项研究试图建立在“群”理论（捕捉并允许我们研究自然界对称性的代数结构）和数学中最著名的未解决问题之一——伽罗瓦反问题——之间出现的迷人联系上。伽罗瓦理论是由法国数学家埃瓦里斯特·伽罗瓦在19世纪发现的，作为研究大于4次的（整数）多项式方程的工具，以及它们何时可以用根式求解。对于这样一个多项式的一组根，伽罗瓦将一个代数结构联系起来，我们现在称之为伽罗瓦群。这个结构是一个集合和一个二元运算，它保留了多项式根的对称性，研究这个运算可以让我们推断根的性质。通过这种方式，伽罗瓦发展了一种理论，人们可以将关于非常复杂的多项式方程的问题转化为关于伽罗瓦群的问题，伽罗瓦群通常更容易、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变为对密码学，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提到的“群”的特殊例子，它们的大小是有限的。因此，所有伽罗瓦群都是有限群，但反过来呢？是否每个有限群都是某个整数多项式的伽罗瓦群？这就是所谓的“逆伽罗瓦问题”（IGP），一个完整的解决方案已经困扰了数学家近200年。一个整数多项式的伽罗瓦群被称为“满足IGP”。一些特殊的情况已经得到了处理（例如，已知大小为n的有限集合的所有对称的群- n次对称群-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宣布了一项研究简单代数群的Weyl群中的IGP的新技术——简单代数群是几何中的一类重要群。这引发了对IGP的新的乐观情绪，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）进一步发展了该技术。本研究计划寻求通过结合上述的群论、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. van der Waerden和J.P. Serre提出的一些长期存在的问题。作为最后一个雄心勃勃的问题，我们寻求在G是最著名和最重要的有限群之一Mathieu群M23的情况下解IGP，在这种情况下IGP是否满足是未知的。

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.