Collaborative Research: Classification of the Finite Simple Groups

摘要-提案0401132 - PI：R.~里昂和R.~ SolomonLyons和所罗门将继续Gorenstein-Lyons-所罗门项目，以创建和发布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。 第一个子计划是对次数至少为9的交错群和无扭李秩至少为3的有限李型群给出一个识别定理的完备集。这些识别定理将与其他卷中获得的局部结构相啮合，通常是一束已知的李或交替类型的拟单群，作为素数阶交换元素的中心化子中的成分而产生。 该项目涉及合作有限几何学家，包括Shpectorov，Gramlich和霍夫曼。第二个项目是分析两类特殊的偶数型有限单群--Klinger-Mason型和中间型。 前者是偶数型群和p-型群，其中p是奇素数，约占散生单群的一半。后一类群粗略地近似于e（G）= 3的群，并包括了特征为2的有限域上定义的BN秩为3的大多数李型群。 这个项目伊萨与Inna Komagina和Kay Magaard合作的。第三个项目是建立偶数型有限单群的不存在性，对于某个奇素数p具有p-唯一子群。这个项目与Gernot Stroth相似。有限群作为离散对象的对称群出现在数学的许多分支中，以及在化学和其他科学中。 具有高度对称性的物体一般都有对称群，这些对称群要么是几乎单群，要么是具有几乎单点群的仿射群。 因此，有限单群，以及关于它们的子群和置换或矩阵表示的问题，在编码理论、晶体学、图论和数论中自然而普遍地存在。 科学家和数学家在研究中认识和运用对称群的能力主要依赖于认识定理，而认识定理几乎都依赖于有限单群的分类定理。 许多这些识别定理已被用于近年来设计强大的计算机软件的有效recognizinggroupsfrom零碎的信息，通常给出了一个生成集的置换或矩阵。 同样，这从根本上依赖于有限单群分类定理的有效性。这个项目，连同其他最近完成的项目和少数广为接受的论文和论文，将提供一个连贯的，可靠的和可读的来源证明这一基本定理。此外，在这个过程中，它还记录了大量关于有限单群的基本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.