Invariable generation in finite groups with applications to algorithmic number theory

这项研究建议位于两个纯数学领域的交界处：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中的对称性的代数结构)中的一个问题和数学中最著名的悬而未决的问题之一：伽罗瓦逆问题之间出现的迷人联系。伽罗瓦理论由法国数学家埃瓦斯特·伽罗瓦在19世纪发现，作为研究次数大于4的(整数)多项式方程的工具，以及何时它们可以用根解。伽罗瓦将一个代数结构与这种多项式的一组根联系在一起，我们现在称之为伽罗瓦群。这种结构是一个集合，加上一个保持多项式根的对称性的二元运算，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，人们可以将关于非常复杂的多项式方程的问题转化为关于它的伽罗瓦群的问题，这通常更容易和更简洁地研究。近年来，该理论与群论和一般数论一起，已从纯粹的学术努力转变为对密码学、电子商务和金融安全的重大贡献。伽罗瓦群是我们在第一段中提到的“群”的特例，它的大小是有限的。因此，所有伽罗瓦群都是有限群，但反过来呢？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“伽罗华逆问题”(IGP)，近200年来，数学家一直没有找到一个完整的解决方案。作为整数多项式的伽罗瓦群的群称为“满足IGP”。已经处理了一些具体的情况(例如，大小为n的有限集的所有对称的群-n次对称群-已知满足IGP)，但即使是一些相对“小”的群也仍然难以捉摸。2008年，数论家Jouve、Cotlaski和Zywina宣布了一种新的方法来研究单代数群--几何中的一类重要群--的Weyl群的IGP。这再次激发了人们对IGP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)进一步发展了这一技术。这项研究计划试图结合上述群论、概率论和组合方法来建立在这些技术的基础上。我们提出的具体问题包括回答关于有限单群(有限群的“积木”)中这些技巧的重要问题，以及回答B.L.van der Wairden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，我们试图解决当G是Mathieu群M23的情况下的IGP-M23是最著名和最重要的有限群之一，其中IGP是否被满足是未知的。

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.