The Classification of the Finite Simple Groups

Abstract

The classification of the finite simple groups was completed sometime during the summer of 1980. To the extent that I can reconstruct things, the last piece in the puzzle was filled in by Ronald Solomon of Ohio State University. At the other chronological extreme, the theory of finite groups can be traced back to its beginnings in the early nineteenth century in the work of Abel, Cauchy, and Galois. Hence the problem of classifying the finite simple groups has a history of over a century and a half. The proof of the Classification Theorem is made up of thousands of pages in various mathematical journals with at least another thousand pages still left to appear in print. Many mathematicians have contributed to the proof; some have spent their entire mathematical lives working on the problem. The problem itself is one of the most natural in mathematics: the group is one of the fundamental structures of modern mathematics; the finite groups are a natural subclass of the class of all groups. Moreover, the finite group theorist is quickly led to consider simple groups via the composition series of a group, and if he is optimistic, to the hope that the finite simple groups might be determined explicitly and much of the structure of the arbitrary finite group retrieved from that of its composition factors. Despite all of this, and despite the fact that most mathematicians learn this much group theory before receiving their Ph.D., the average mathematician does not seem to known much about the classification problem or the mathematics developed to solve it. Within the obvious space limitations of this article, I hope to convey some idea of how the finite simple groups are classified and to relate some of the history of the effort. A more complete description appears in [6], while a very detailed two volume account (by Daniel Gorenstein) is in preparation. A preliminary version of the first quarter of Gorenstein's work appears in [19]. The proceedings of two recent conferences on simple groups containing expository articles on the classification will soon appear in [12] and [13]. Finally an article by Walter Feit on the history of finite group theory through 1961 will appear in [14]. I have included a reasonably lengthy bibliography. Still, many important papers are omitted as they are not directly encountered in the brief outline provided. Other fundamental papers have yet to appear. More complete bibliographies are contained in some of the books mentioned above. Section 1. The Finite Simple Groups