Collaborative Research: Classification of the Finite Simple Groups

摘要--提案0401132--圆周率：R.莱昂斯和R.所罗门将继续戈伦斯坦-莱昂斯-所罗门项目，创建并出版有限单群分类的第二代证明。该项目的其余部分被细分为三个主要的子项目。第一个子项目是为至少9次的交错次数群和至少3次的李型李型有限群提供一套完整的识别定理。这些识别定理将与在其他卷中获得的局部结构相吻合，这些卷通常是作为素数阶交换元的中心子中的分量而出现的已知李型或交替型拟单群的花束。这个项目涉及与有限几何学家的合作，包括Shectorov，Gramlich和Hoffman。第二个项目是分析两类特殊的偶型有限单群--Klinger-Mason型群和中间型群。对于某些奇素数p，前者被刻画为事件型群和p型群，并且包括大约一半的零星单群。后者粗略地逼近了e(G)=3的群，包括了特征为2的有限域上定义的BN-秩3Lie型群的大部分。这是Inna Korchagina和Kay Magaard的合作项目。第三个项目是建立对某个奇素数p具有p-唯一子群的偶型有限单群的不存在性。这个项目是与Gernot Stroth合作的。有限群作为离散对象的对称群出现在数学的许多分支中，在化学和其他科学中也是如此。具有高度对称性的物体通常具有对称群，其或者是几乎简单的群，或者是具有几乎简单的点群的仿射群。因此，有限单群及其子群和置换或矩阵表示的问题在编码论、结晶学、图论和数论中自然地和普遍地出现。科学家和数学家对他们研究中出现的对称群的理解和利用的能力主要取决于识别定理，而识别定理最终几乎都依赖于有限单群的分类定理。近年来，许多这样的识别定理被用来设计功能强大的计算机软件，用于从通常由一组生成的排列或矩阵给出的零碎信息中有效地识别群。同样，这从根本上依赖于有限单群的分类定理的有效性。这个项目，连同最近完成的其他项目和一些广受欢迎的论文和论文，将为这一基本定理的证明提供一个连贯的、可靠的和可读的来源。此外，在这个过程中，它还记录了关于有限单群的丰富的认识定理和详细的子群信息。

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.