Collaborative Research -- Classification of the Finite Simple Groups

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ABSTRACT -- PROPOSAL 0401132 -- PI's: R.~Lyons and R.~SolomonLyons and Solomon will continue the Gorenstein-Lyons-Solomon project tocreate and publish a second-generation proof of the Classification of theFinite Simple Groups. The remainder of the project is subdivided intothree major subprojects. The first subproject is to provide a completeset of recognition theorems for the alternating groups of degree at leastnine and for the finite groups of Lie type of untwisted Lie rank at leastthree. These recognition theorems will mesh with the local structureobtained in other volumes, generally a bouquet of known quasisimple groupsof Lie or alternating type, arising as components in the centralizers ofcommuting elements of prime order. This project involves collaborationwith finite geometers, including Shpectorov, Gramlich and Hoffman. Thesecond project is to provide an analysis of two special classes of finitesimple groups of even type -- those of Klinger-Mason type and those ofintermediate type. The former are characterized as groups of both eventype and p-type for some odd prime p, and include approximately half ofthe sporadic simple groups. The latter class roughly approximates thegroups with e(G) = 3 and includes most of the groups of Lie type ofBN-rank 3 defined over finite fields of characteristic 2. This project isa collaboration with Inna Korchagina and Kay Magaard. The third project isto establish the non-existence of finite simple groups of even type havinga p-uniqueness subgroup for some odd prime p. This project is acollaboration with Gernot Stroth.Finite groups arise as the symmetry groups of discrete objects in manybranches of mathematics, as well as in chemistry and other sciences. Objects having a high degree of symmetry generally have symmetry groupswhich are either almost simple groups or affine groups having almostsimple point groups. As such, finite simple groups, and questions abouttheir subgroups and representations by permutations or matrices, arisenaturally and pervasively in coding theory, crystallography, graph theoryand number theory. The ability of scientists and mathematicians tounderstand and use the symmetry groups which arise in their researchdepends critically on recognition theorems, almost all of which relyeventually on the classification theorem of the finite simple groups. Many of these recognition theorems have been used in recent years todesign powerful computer software for the efficient recognition of groupsfrom fragmentary information, usually given by a generating set ofpermutations or matrices. Again this relies fundamentally on the validityof the classification theorem of the finite simple groups. This project,in conjunction with other recently completed projects and a small numberof well-accepted treatises and papers, will provide a coherent, reliableand readable source for the proof of this fundamental theorem. Inaddition, in the process, it is documenting a wealth of recognitiontheorems and detailed subgroup information about the finite simple groups.