# Classification of the Finite Simple Groups

When the Classification of Finite Simple Groups was completed by Michael Aschbacher and Stephen Smith in 2004, Aschbacher wrote the article 'The Status of the Classification of the Finite Simple Groups', Notices of the American Mathematical Society 51 (7) (2004), 736-740. This beautifully written article should be understandable by anyone who has done an undergraduate group theory course. We give a brief extract.

The Classification of Finite Simple Groups, by Michael Aschbacher.

Common wisdom has it that the theorem classifying the finite simple groups was proved around 1980. However, the proof of the Classification is not an ordinary proof because of its length and complexity, and even in the eighties it was a bit controversial. Soon after the theorem was established, Gorenstein, Lyons, and Solomon (GLS) launched a program to simplify large parts of the proof and, perhaps of more importance, to write it down clearly and carefully in one place, appealing only to a few elementary texts on finite and algebraic groups and supplying proofs of any "well-known" results used in the original proof, since such proofs were scattered throughout the literature or, worse, did not even appear in the literature. However, the GLS program is not yet complete, and over the last twenty years gaps have been discovered in the original proof of the Classification. Most of these gaps were quickly eliminated, but one presented serious difficulties. The serious gap has recently been closed, so it is perhaps a good time to review the status of the Classification.
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One can conceive of an analysis of the finite groups based on a solution to the following two problems:

The Classification Problem. Determine all finite simple groups.

The Extension Problem. Given groups $X$ and $Y$, determine all extensions of $X$ by $Y$; i.e. determine all groups $G$ with a normal subgroup $H$ such that $H \cong X$ and $G/H \cong Y$.

In practice the Extension Problem is too hard, except in special cases. It seems better not to look too closely at the general finite group, but instead when faced with a problem about finite groups, to attempt to reduce the problem or a related problem to a question about simple groups or groups closely related to simple groups. Then using the Classification of the finite simple groups and knowledge of the simple groups, solve the reduced problem. Note this procedure works only if one knows enough about simple groups to solve the problem for simple groups; this is where the Classification comes in: it supplies an explicit list of groups which can be studied in detail using the effective description of the groups supplied by the Classification.
This approach to solving group theoretic problems has been in use since about 1980, when the finite simple groups were deemed to have been classified. It has been extremely successful: virtually none of the major problems in finite group theory that were open before 1980 remain open today. Moreover, finite group theory has been used to solve problems in many branches of mathematics.

In short, the Classification is the most important result in finite group theory, and it has become increasingly important in other areas of mathematics.

Last Updated March 2024