ABSTRACT -- PROPOSAL 0401132 -- PI's: R.~Lyons and R.~Solomon
Lyons and Solomon will continue the Gorenstein-Lyons-Solomon project to create and publish a second-generation proof of the Classification of the Finite Simple Groups. The remainder of the project is subdivided into three major subprojects. The first subproject is to provide a complete set of recognition theorems for the alternating groups of degree at least nine and for the finite groups of Lie type of untwisted Lie rank at least three. These recognition theorems will mesh with the local structure obtained in other volumes, generally a bouquet of known quasisimple groups of Lie or alternating type, arising as components in the centralizers of commuting elements of prime order. This project involves collaboration with finite geometers, including Shpectorov, Gramlich and Hoffman. The second project is to provide an analysis of two special classes of finite simple groups of even type -- those of Klinger-Mason type and those of intermediate type. The former are characterized as groups of both even type and p-type for some odd prime p, and include approximately half of the sporadic simple groups. The latter class roughly approximates the groups with e(G) = 3 and includes most of the groups of Lie type of BN-rank 3 defined over finite fields of characteristic 2. This project is a collaboration with Inna Korchagina and Kay Magaard. The third project is to establish the non-existence of finite simple groups of even type having a p-uniqueness subgroup for some odd prime p. This project is a collaboration with Gernot Stroth.
Finite groups arise as the symmetry groups of discrete objects in many branches of mathematics, as well as in chemistry and other sciences. Objects having a high degree of symmetry generally have symmetry groups which are either almost simple groups or affine groups having almost simple point groups. As such, finite simple groups, and questions about their subgroups and representations by permutations or matrices, arise naturally and pervasively in coding theory, crystallography, graph theory and number theory. The ability of scientists and mathematicians to understand and use the symmetry groups which arise in their research depends critically on recognition theorems, almost all of which rely eventually on the classification theorem of the finite simple groups. Many of these recognition theorems have been used in recent years to design powerful computer software for the efficient recognition of groups from fragmentary information, usually given by a generating set of permutations or matrices. Again this relies fundamentally on the validity of the classification theorem of the finite simple groups. This project, in conjunction with other recently completed projects and a small number of well-accepted treatises and papers, will provide a coherent, reliable and readable source for the proof of this fundamental theorem. In addition, in the process, it is documenting a wealth of recognition theorems and detailed subgroup information about the finite simple groups.
