On the product decomposition conjecture for finite simple groups

关于有限单群的乘积分解猜想

• 批准号: EP/N010957/1
• 负责人: Nick Gill
• 金额: $ 10.07万
• 依托单位: University of South Wales
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FN010957%2F1
• 关键词: product; decomposition; conjecture; finite; simple

Within mathematics the study of symmetry is called "group theory". Given some kind of object (physical or mathematical), its "symmetry group" is the set of transformations of the object that preserve its structure. A very symmetrical object will have a large symmetry group, an asymmetrical object will have a tiny symmetry group. The cube, for instance, has a symmetry group of size 48 - these are all the reflections and rotations of 3-dimensional space that leave the vertices of the cube unchanged set-wise. Given such a group of symmetries we can consider the "composition" of two group elements and it is clear that such a composition will itself be a group element. For instance if I rotate the cube around one axis, and then again around another, the end result will be the same as if I had rotated the cube around a third axis.This project studies groups of a particular type. Firstly, they are FINITE; secondly, they are SIMPLE. In this context, simple means that the group cannot be "broken up" into smaller pieces. It is important to note that simple does not mean easy! The study of the finite simple groups is an extraordinarily rich area of mathematics containing many very difficult open questions.This research starts with the following set-up: Suppose that we have a finite simple group G and a subset A inside G with A of size at least 2. It is well-known that any element of G can be written as a composition of some number N of elements "of the same type" as A. (Here "of the same type" has a technical meaning that we won't discuss. Roughly speaking though, if one looks at the cube example, one can see that a ROTATION has different qualities to a REFLECTION. The idea of "type" is a refinement of this qualitative distinction.)We would like to write all of the elements of G in the most efficient way possible using elements of the same type as A. By efficient we mean using as few compositions as possible. The Product Decomposition Conjecture (PDC) asserts that elements of finite simple groups can be written very efficiently indeed. APPLICATIONS: Although the setting for this research is very abstract, there are a surprising number of rather concrete applications. One of the original motivations for the PDC, for instance, was in the explicit construction of EXPANDER FAMILIES. These are mathematical models of efficient networks which have a myriad of applications in mathematics, computer science and elsewhere. It turns out that one can use notions of "efficiency" in finite simple groups to construct expander families.METHODS: The primary tool at our disposal to prove PDC is the Classification of Finite Simple Groups (CFSG). This monumental theorem was proved by hundreds of mathematicians over a period of about 40 years, culminating in 2001. CFSG asserts that all finite simple groups are on an explicit (infinitely long) list. Thus to prove PDC it is enough to prove the result for all of the groups on the list. In fact some of the groups on the list have already been attended to in earlier collaborative work of the Principal Investigator and others.It is expected that research into the PDC on the groups that remain will, in addition to yielding a proof of PDC, shed light on some of the deep and mysterious properties of the finite simple groups.

在数学中，对对称性的研究被称为“群论”。给定某种对象（物理的或数学的），它的“对称群”是保持其结构的对象的变换的集合。一个非常对称的物体会有一个大的对称群，一个不对称的物体会有一个小的对称群。例如，立方体有一个大小为48的对称群--这些是三维空间的所有反射和旋转，使立方体的顶点保持不变。给定这样一个对称群，我们可以考虑两个群元素的“合成”，很明显，这样的合成本身就是一个群元素。例如，如果我绕一个轴旋转立方体，然后再绕另一个轴旋转，最终结果将与绕第三个轴旋转立方体相同。这个项目研究特定类型的群体。首先，它们是有限的;其次，它们是简单的。在这种情况下，简单意味着群体不能被“分解”成更小的部分。值得注意的是，简单并不意味着容易！有限单群的研究是一个非常丰富的数学领域，包含着许多非常困难的开放性问题，本研究从以下的设置开始：假设我们有一个有限单群G和G内的一个子集A，A的大小至少为2。众所周知，G的任何元素都可以写成与A“同类型”的N个元素的组合。(Here“同一类型”有一个技术上的含义，我们不会讨论。粗略地说，如果你看一下立方体的例子，你可以看到旋转和反射有不同的性质。“类型”的概念是对这种质的区分的细化。）我们希望使用与A相同类型的元素以最有效的方式编写G的所有元素。高效是指使用尽可能少的成分。乘积分解猜想（Product Decomposition Conjecture，PDC）认为有限单群的元素可以非常有效地写出。应用：虽然这项研究的背景非常抽象，但有大量令人惊讶的具体应用。例如，PDC最初的动机之一就是明确地构建扩展家庭。这些是有效网络的数学模型，在数学、计算机科学和其他领域有着无数的应用。事实证明，人们可以使用有限简单群中的“效率”概念来构建扩展族。METHODS：在我们的处置证明PDC的主要工具是有限简单群的分类（CFSG）。这个不朽的定理被数百名数学家证明了大约40年的时间，最终在2001年达到顶峰。CFSG断言所有有限单群都在一个显式（无限长）列表上。因此，为了证明PDC，证明列表上所有组的结果就足够了。事实上，名单上的一些群体已经参加了早期的合作工作的主要研究者和其他人。预计研究到PDC的团体，仍然将，除了产生一个证明PDC，揭示了一些深刻的和神秘的性质的有限简单的群体。

项目成果

Conway groupoids and completely transitive codes

康威群群和完全传递码

• DOI: 10.48550/arxiv.1410.4785
• 发表时间: 2014
• 期刊: arXiv e-prints
• 作者: Gill Nick

QUASIRANDOM GROUP ACTIONS

拟随机群行动

• DOI: 10.1017/fms.2016.8
• 发表时间: 2016
• 期刊: Forum of Mathematics, Sigma
• 作者: GILL N

Conway's groupoid and its relatives

康威群胚及其亲属

• DOI: 10.48550/arxiv.1604.04429
• 发表时间: 2016
• 期刊: arXiv e-prints
• 作者: Gill Nick

Abelian covers of alternating groups

交替群的阿贝尔覆盖

• DOI: 10.1007/s00013-016-0926-y
• 发表时间: 2016
• 期刊: Archiv der Mathematik
• 影响因子: 0.6
• 作者: Barrantes D

Generating groups using hypergraphs

• DOI: 10.1093/qmath/haw001
• 发表时间: 2014-05
• 期刊: arXiv: Group Theory
• 作者: Nick Gill;Neil I. Gillespie;A. Nixon;Jason Semeraro