LYONS/SOLOMON 97-01253 This project is a continuation of the revision project begun with Danny Gorenstein for the proof of the classification of the finite simple groups. The proposers contemplate publishing three volumes during the period of the grant(two already having been published). The first of the three is devoted to laying the groundwork for establishing the many properties of the known simple groups necessary for the proof. The second and third are devoted to various uniqueness theorems, the key underlying theorems for the proof. In the second will be Bender's strongly embedded subgroup theorem and variations; new theorems on p-component uniqueness subgroups; and a version of the global C(G,T)-theorem, partly in collaboration with Richard Foote. In the third will be preparation theorems for the uniqueness case for groups of even type, connecting p-uniqueness and 2- uniqueness case from an amalgam-theoretic point of view, written by Gernot Stroth. The finite simple groups were classified in an ever-accelerating process which culminated around 1980, in a proof depending on hundreds of articles totaling tens of thousands of pages. Because of the central position of this result in group theory and its many applications, as well as the length and complexity of the proof, the current effort was begun. The goal is a coherent proof, as direct and clear as possible and using the best ideas at present. The proof is now conceived as a series of roughly ten volumes, of which the first two have already appeared.
