Uncovering the Subgroup Structure of E8

一个群就是一组对象在它们之间进行排列的方式的集合，这样集合中任何排列的相反排列（逆）也在集合中，而且如果你在集合中选择两个排列，然后做一个，那个排列仍然在集合中。这样的群的标准例子包括一些对象（如正方形）的点的排列群（八对称），魔方（2125922464947725402112000对称）或圆（无穷多个对称）。子群仅仅是一个子集，它也是一个群，最大子群是指除了群本身之外，不包含在任何子群中的子群。理解有限群的极大子群等价于理解有限群可以置换一组对象的所有方式，所以极大子群在任何有一组可以移动的事物的地方都有各种各样的应用。例子包括角落的一个物理对象和解决方案的一个方程。在1985年，Aschbacher和斯科特证明，所有极大子群的所有有限群可以理解，如果一个人可以解决两个问题，其中之一是了解所有极大子群的一个小类的有限群，所谓的“几乎简单”。这些只是在一个长达数十年的项目中被分类，需要数千页。理解它们的极大子群还需要几十年的时间，我们仍然远远没有完全理解，如果这样的事情是可能的话。有限单群分为四个家族：交替，经典，例外和零星。要理解交替群和经典群的极大子群，需要理解阶数较小的简单群，因此有一个递归算法是可能的，但可能没有简单的答案。有26个零星群，其中25个群的所有极大子群都是已知的，只有少数26个群缺失。对于特殊群，有八种类型的群，写作G2，2G2，F4，2F4，E6，2E6，E7和E8。群G2、2G2和2F4都很小，它们的最大子群在1990年左右被理解。重要的工作李贝克和塞茨在20世纪90年代末和21世纪初了一个很好的描述了许多极大子群，让我们在相同的立场，为交替和经典的群体。它把我们简化到子群也是简单子群的情况，所以我们需要理解简单子群。他们给出了一个可能是最大的简单子群的列表，范围从F4的几十个到E8的几百个。有了这些信息，我在2020年设法给出了F4、E6和2E6型群的最大子群的完整分类。一年后，我几乎完全分类了E7的极大子群。但E8远大于E7，而用于较小群的方法在E8上变得不切实际。本项目旨在改进用于较小群的方法和算法，使E8的极大子群可以像其他群一样分类。这将结束一个项目跨越几十年和几千页的数学。

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.