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 decadeslong 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.
