Collaborative Research: Classification of the Finite Simple Groups

合作研究：有限简单群的分类

• 批准号: 0400533
• 负责人: Ronald Solomon
• 金额: $ 13.5万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400533&HistoricalAwards=false
• 关键词: Collaborative; Research; Classification; Finite; Simple

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.

摘要 - 提案0401132-PI：R。〜Lyons和R.〜所罗门林和所罗门将继续进行Gorenstein-Lyons-Solomon项目Togreate并发布第二代简单组的分类证明。该项目的其余部分被细分为主要的主要子标。 第一个子项目是为交替的学位组提供一个完整的识别定理，至少是lie type的有限型组，至少是谎言类型的有限组。这些识别定理将与其他体积中的局部结构进行融为一体，通常是一束已知的Quasisimple群体的花束或交替类型的一束，作为质量顺序的元素的中心化元素中的组成部分。 该项目涉及合作与有限几何图形，包括Sheptorov，Gramlich和Hoffman。第三个项目将对两种类型的有限类型组（Klinger-Mason类型和中等类型的类型）提供分析。 前者的特征是某些奇数P的Eventype和P型组的组，其中大约一半的零星简单组。后类大约近似于E（g）= 3的组，其中包括在特征2的有限领域定义的大多数Lie类型的Ofbn-of ofbn-of-of of of of of bn-rank 3。第三个项目是为某些奇数p的有限简单组建立有限的简单组，即具有p-唯一性亚组的不存在。 该项目是与Gernot Stroth一起进行的。限制群体是许多数学分布以及化学和其他科学中的离散对象的对称组。 具有高度对称性的对象通常具有对称性组的群体，要么几乎是简单的群体，要么是具有几乎简单点基组的仿射组。 因此，在编码理论，晶体学，图理论和数字理论中，有限的简单群体以及通过排列或矩阵对您的亚组和表示的问题。 科学家和数学家的能力toudsterstand并使用其研究中出现的对称群体依赖于识别定理，几乎所有这些都依赖于有限简单组的分类定理。 近年来，这些识别定理中的许多已使用，用于有效识别片段信息的组，通常由生成的persmutations或矩阵提供。 同样，这从根本上依赖于有限简单组的分类定理的有效性。该项目与其他最近完成的项目以及少量被接受的论文和论文结合在一起，将提供一个连贯的，可靠的可读来源，以证明该基本定理。在此过程中，它正在记录大量的识别理论和有关有限简单组的详细子组信息。