Ebook: The Status of the Classification of the Finite Simple Groups

USA.: California Institute of Technology [Amer. Math. Soc, 51 (2004), No. 7, 736-740, eBook, English]Abstract
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. I will begin slowly with an introduction to the problem and with some motivation.Contents
Jordan-Hölder Theorem
Example
The Classification Problem
The Extension Problem
Classification Theorem
Step 1
Step 2
Case I
Case II
Classification of QTKE Groups
References