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| title | chunk | source | category | tags | date_saved | instance |
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| Classification of finite simple groups | 3/4 | https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups | reference | science, encyclopedia | 2026-05-05T09:08:19.935542+00:00 | kb-cron |
== Second-generation classification == The proof of the theorem, as it stood around 1985 or so, can be called first generation. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called a second-generation classification proof. This effort, called "revisionism", was originally led by Daniel Gorenstein, and coauthored with Richard Lyons and Ronald Solomon. As of 2023, ten volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005, 2018a, 2018b; & Inna Capdeboscq, 2021, 2023). In 2012 Solomon estimated that the project would need another 5 volumes, but said that progress on them was slow. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from the second generation proof being written in a more relaxed style.) However, with the publication of volume 9 of the GLS series, and including the Aschbacher–Smith contribution, this estimate was already reached, with several more volumes still in preparation (the rest of what was originally intended for volume 9, plus projected volumes 10 and 11). Aschbacher and Smith wrote their two volumes devoted to the quasithin case in such a way that those volumes can be part of the second generation proof. Gorenstein and his collaborators have given several reasons why a simpler proof is possible.
The most important thing is that the correct, final statement of the theorem is now known. Simpler techniques can be applied that are known to be adequate for the types of groups we know to be finite simple. In contrast, those who worked on the first generation proof did not know how many sporadic groups there were, and in fact some of the sporadic groups (e.g., the Janko groups) were discovered while proving other cases of the classification theorem. As a result, many of the pieces of the theorem were proved using techniques that were overly general. Because the conclusion was unknown, the first generation proof consists of many stand-alone theorems, dealing with important special cases. Much of the work of proving these theorems was devoted to the analysis of numerous special cases. Given a larger, orchestrated proof, dealing with many of these special cases can be postponed until the most powerful assumptions can be applied. The price paid under this revised strategy is that these first generation theorems no longer have comparatively short proofs, but instead rely on the complete classification. Many first generation theorems overlap, and so divide the possible cases in inefficient ways. As a result, families and subfamilies of finite simple groups were identified multiple times. The revised proof eliminates these redundancies by relying on a different subdivision of cases. Finite group theorists have more experience at this sort of exercise, and have new techniques at their disposal. Aschbacher (2004) has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, a third generation program. One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.
=== Length of proof === Gorenstein has discussed some of the reasons why there might not be a short proof of the classification similar to the classification of compact Lie groups.
The most obvious reason is that the list of simple groups is quite complicated: with 26 sporadic groups there are likely to be many special cases that have to be considered in any proof. So far no one has yet found a clean uniform description of the finite simple groups similar to the parameterization of the compact Lie groups by Dynkin diagrams. Atiyah and others have suggested that the classification ought to be simplified by constructing some geometric object that the groups act on and then classifying these geometric structures. The problem is that no one has been able to suggest an easy way to find such a geometric structure associated with a simple group. In some sense, the classification does work by finding geometric structures such as BN-pairs, but this only comes at the end of a very long and difficult analysis of the structure of a finite simple group. Another suggestion for simplifying the proof is to make greater use of representation theory. The problem here is that representation theory seems to require very tight control over the subgroups of a group in order to work well. For groups of small rank, one has such control and representation theory works very well, but for groups of larger rank no-one has succeeded in using it to simplify the classification. In the early days of the classification, there was a considerable effort made to use representation theory, but this never achieved much success in the higher rank case.
== Consequences of the classification == This section lists some results that have been proved using the classification of finite simple groups.
A breakthrough in the best known theoretical algorithm for the graph isomorphism problem in 1982 The Schreier conjecture The Signalizer functor theorem The B conjecture The Schur–Zassenhaus theorem for all groups (though this only uses the Feit–Thompson theorem). A transitive permutation group on a finite set with more than 1 element has a fixed-point-free element of prime power order. The classification of 2-transitive permutation groups. The classification of rank 3 permutation groups. The Sims conjecture Frobenius's conjecture on the number of solutions of xn = 1. Non-abelian finite simple groups are characterized by their commuting graphs.
== See also == O'Nan–Scott theorem
== Citations ==