On the product decomposition conjecture for 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集。一个非常对称的物体会有一个很大的对称群，一个不对称的物体会有一个很小的对称群。例如，立方体有一个大小为48的对称组-这些是三维空间的所有反射和旋转，使立方体的顶点在集合方向保持不变。给出这样一组对称性，我们可以考虑两个群元素的“组成”，很明显，这样的组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，最终结果将与我绕第三个轴旋转立方体相同。这个项目研究特定类型的组。首先，它们是有限的；其次，它们是简单的。在这种情况下，Simple意味着不能将群体“分解”成更小的碎片。重要的是要注意，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，其中包含许多非常困难的开放问题。这项研究从以下设置开始：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相同类型的N个元素的组合。(在这里，“相同类型”有一个我们不讨论的技术含义。不过，粗略地说，如果看一下立方体的例子，就会发现旋转与反射具有不同的性质。“类型”的概念是这种定性区别的细化。)我们希望以尽可能有效的方式使用与A相同类型的元素来编写G的所有元素。乘积分解猜想(PDC)断言有限单群的元素确实可以非常有效地写出来。应用：虽然这项研究的背景非常抽象，但有数量惊人的相当具体的应用。例如，PDC的最初动机之一是显式地构建扩张器家族。这些是高效网络的数学模型，在数学、计算机科学和其他领域有无数的应用。证明了有限单群的“有效性”概念可以用来构造扩张族。方法：证明PDC的主要工具是有限单群的分类。在大约40年的时间里，数百名数学家证明了这一不朽的定理，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，要证明PDC，只需证明列表上所有组的结果就足够了。事实上，名单上的一些群已经在首席调查员和其他人的早期合作工作中得到了关注。预计对剩余群的PDC的研究，除了产生PDC的证明外，还将揭示有限单群的一些深刻和神秘的性质。

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.