Algebraic group techniques for finite(ly presented) groups

有限（已知）群的代数群技术

• 批准号: 123863751
• 负责人: Professor Dr. Wilhelm Plesken
• 金额: --
• 依托单位: Lehrstuhl für Algebra und Darstellungstheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2011-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/123863751?language=en
• 关键词: Algebraic; group; techniques; finite; ly

1.) In the first part of the project we discovered that finite matrix groups G that are defined, as algebraic groups, over the rational field can be studied via modules over a twisted group ring constructed from the field and the semidirect product of G with the Galois group. Motivated by the fact that it is possible to define characters for those twisted group rings, it remains to study which parts of the classical character theory can be adapted. The theory of Schur indices, especially the theorem of Brauer-Witt, will be in the center of interest.2.) In the second part of the project, the efficient methods to compute the epimorphisms of a finitely presented group onto all finite groups of type L2 have been extended to types L3 and U3. The theoretical insights gained from this are to be developed into a general character theory, which can be applied for instance to determine the Aschbacher class of a matrix group over a finite field. Both, finite group presentations and finitely generated matrix groups over rings of algebraic integers, are to be investigated by the developed methods. E.g., results by Lubotzky on subgroups of SL(n, Z) are to be made constructive.

1.）的人。在该项目的第一部分，我们发现，有限矩阵群G的定义，作为代数群，在合理的领域可以研究通过模块的扭曲群环从外地和半直积的G与伽罗瓦群。由于有可能定义这些扭群环的特征标，因此还需要研究经典特征标理论的哪些部分可以适用。Schur指数的理论，特别是Brauer-Witt定理，将成为兴趣的中心。在项目的第二部分中，将计算一个可表示群到所有L2型有限群上的满态的有效方法推广到了L3和U3型。从这一点获得的理论见解将发展成为一个一般的性质理论，它可以应用于例如确定Aschbacher类的矩阵群在有限领域。这两个，有限群介绍和代数整数环上的矩阵群，是由发达国家的方法进行调查。例如，在一个示例中，Lubotzky关于SL（n，Z）子群的结果是建设性的。