The Classification of Finite Simple Groups

My aim in this lecture will be to try to convince you that the classification of the finite simple groups is nearing its end. This is, of course, a presumptuous statement, since one does not normally announce theorems as "almost proved". But the classification of simple groups is unlike any other single theorem in the history of mathematics, since the final proof will cover at least 5,000 journal pages. Moreover, at the present time, perhaps 80% of these pages exist either in print or in preprint form. One obtains a better perspective of the subject if instead of thinking of the classification as a single theorem, one views it as an entire field of mathematics—the structure of finite groups. Then when I say that there are some 4,000 pages in print, proving many general and specific results about simple groups, it should sound entirely reasonable, since one can make the same claim concerning many areas of mathematics. Thus my task is really to convince you that we have established so many results about simple groups and have developed sufficient techniques for completing the classification. There are other reasons for skepticism besides my premature announcement of the impending completion of the classification. Indeed, to the nonspecialist, simple group theory appears to be in a rather chaotic state. Strange sporadic simple groups dot the landscape—26 at last count; and they appear to be widely unrelated to each other. The five Mathieu groups, 100 years old, examples of highly transitive permutation groups, the four groups of Janko, each arising from the study of centralizers of involutions, the three Conway groups, determined from the automorphisms of a certain integral lattice in 24-dimensional Euclidean space, etc. And now there comes the Fischer-Griess monster, of order over 10; to be precise: