Uncovering the Subgroup Structure of E8

组仅仅是在它们之间排列一组对象的方法的集合，使得集合中任何排列的相反排列(相反的排列)也在集合中，并且如果您在集合中选择两个排列，并且一个接着另一个，则该排列仍然在集合中。这类群的标准例子包括某些对象的点的排列的群，例如正方形(8个对称)、魔方(2125922464947725402112000个对称)或圆(无限多个对称)。子群是也是群的子集，极大子群是除了群本身之外不包含在任何子群中的子群。理解有限群的极大子群等同于理解有限群可以置换一组对象的所有方法，因此极大子群在任何地方都有各种各样的应用，只要有一组可以移动的东西。1985年，Aschbacher和Scott证明，如果一个人能解决两个问题，其中一个问题是理解一小类有限群的所有极大子群，则可以理解所有有限群的所有极大子群。这些刚刚被归类在一个长达数十年、耗时数千页的项目中。了解它们的极大子群还需要几十年的时间，如果有可能的话，我们还远未完全了解。有限的简单群分为四个族：交替的、经典的、例外的和零星的。要理解交替群和经典群的极大子群，需要理解较小阶数的简单群，因此有可能有一个递归算法，但可能没有简单的答案。有26个零星群，其中25个已知所有极大子群，只有几个在第26个丢失。对于特殊组，有八种类型的组，书面G2、2G2、F4、2F4、E6、2E6、E7和E8。G2、2G2和2F4群都很小，它们的最大亚群大约在1990年被理解。Liebeck和Seitz在20世纪90年代末和21世纪初的重要工作很好地刻画了许多极大子群，使我们处于与交错群和经典群相同的位置。它将我们简化到子群也是简单的情况，所以我们需要理解简单的子群。他们列出了一张可能是最大的简单子群的清单，范围从F4的几十个到E8的几百个。有了这些信息，我在2020年设法对F4、E6和2E6型群的最大子群进行了完整的分类。一年后，我也几乎完全地对E7的极大子群进行了分类。但E8比E7大得多，用于较小群的相同技术对E8来说变得不实用。本项目旨在改进用于较小群的方法和算法，以便E8的最大子群可以像对其他群一样被分类。这将结束一个跨越几十年和数千页数学的项目。

A group is simply a collection of ways of permuting a set of objects amongst themselves, such that the opposite permutation (the inverse) of any permutation in the collection is also in the collection, and also if you choose two permutations in the collection, and do one then the other, that permutation is still in the collection. Standard examples of such groups include groups of permutations of the points of some object, such as a square (eight symmetries), Rubik's cube (2125922464947725402112000 symmetries) or a circle (infinitely many symmetries).A subgroup is simply a subset that is also a group, and a maximal subgroup is a subgroup that isn't contained in any subgroup other than the group itself. Understanding the maximal subgroups of finite groups is equivalent to understanding all ways that finite groups can permute a set of objects, so maximal subgroups have a variety of applications wherever there is a set of things that can be moved around. Examples include the corners of a physical object and solutions to an equation.In 1985, Aschbacher and Scott proved that all maximal subgroups of all finite groups could be understood if one could solve two problems, one of which was understanding all maximal subgroups of a small class of finite groups, called 'almost simple'. These had just been classified in a decades-long project taking thousands of pages. Understanding their maximal subgroups would take decades more, and we are still far from a complete understanding, if such a thing is even possible.The finite simple groups split into four families: alternating, classical, exceptional, and sporadic. To understand the maximal subgroups of alternating and classical groups requires understanding simple groups of smaller order, so there is a recursive algorithm possible, but likely no simple answer. There are 26 sporadic groups, and all maximal subgroups are known for 25 of them, with only a few missing for the 26th. For exceptional groups, there are eight types of groups, written G2, 2G2, F4, 2F4, E6, 2E6, E7 and E8. The groups G2, 2G2 and 2F4 are all small, and their maximal subgroups were understood by around 1990. Important work of Liebeck and Seitz in the late 1990s and early 2000s gave a good description of many of the maximal subgroups, leaving us in the same position as for alternating and classical groups. It reduced us to the case where the subgroup is also simple, so we needed to understand simple subgroups. They produced a list of the possible simple subgroups that could be maximal, which can range from a few dozen for F4 to several hundred for E8.Armed with this information, in 2020 I managed to produce a complete classification of the maximal subgroups of groups of types F4, E6 and 2E6. A year later I almost completely classified the maximal subgroups of E7 as well. But E8 is far larger than E7, and the same techniques that were used for the smaller groups become impractical for E8.This project aims to improve the methods and algorithms used for the smaller groups, so that the maximal subgroups of E8 can be classified just as for the other groups. This would bring to a close a project spanning several decades and several thousands of pages of mathematics.