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# Classification of finite simple groups

The classification of the finite simple groups, also called the enormous theorem, is believed to classify all finite simple groups. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic building blocks of the natural numbers. The Jordan-Hölder theorem is a more precise way of stating this fact about finite groups.

The "theorem" is mainly a convenient way of describing a vast body of mathematical writing, made up of tens of thousands of pages in 500-odd journal articles written by about 100 authors, published mostly between 1955 and 1983. There has been some controversy as to whether the resulting proof is complete and correct, given its length and complexity.

## The classification theorem

Theorem. Every finite simple group is one of 26 sporadic simple groups or belongs (up to isomorphism) to at least one of the following three infinite families:

Some consider the Tits group a 27th sporadic group because it is not strictly a group of Lie type, but this difference has no impact on the classification theorem.

The first sporadic groups to be discovered were the five Mathieu groups, discovered in the 1860s by Emile Mathieu. The other 21 sporadic groups were found between 1965 and 1975. 20 of the 26 sporadic groups form three families (one of which is the family of Mathieu groups), and are subgroups or subquotients of the Monster group, the sporadic group with the largest order. The remaining six sporadic groups defy classification and are called "pariahs."

The classification theorem has widespread applications in many branches of mathematics, as questions about the structure of finite groups (and their action on other mathematical objects) can often be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can often be answered by examining only finitely many configurations. In particular, each of the infinite families can often be disposed of by a single argument.

Some doubts remain as to whether a proof spread over 500-odd articles is complete and correct, and these doubts were justified to a considerable extent when gaps in the argument were found, although all gaps known to date have been fixed. Parts of the alleged proof also remain unpublished. Jean-Pierre Serre is a notable skeptic about the claimed proof of the enormous theorem.

For over a decade, experts knew of a "serious gap" (according to Michael Aschbacher) in Geoff Mason's unpublished classification of quasithin groups. Daniel Gorenstein's 1983 announcement that the finite simple groups had all been classified was partly based on his belief that the quasithin case was finished. Aschbacher filled this gap in the early 90s, in work that remains unpublished. Aschbacher and Steve Smith have published a different proof for the quasithin case, one running about 1300 pages and filling two volumes.

## 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.

As of 2005, six volumes of the second generation proof have been published, with most of the balance of the proof in manuscript. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from second generation proof being written in a more relaxed style.) 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 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 subfamiles 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.

## Third-generation classification

Some designate the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, as a third generation proof.