Algebraic group techniques for finite(ly presented) groups

1)在项目的第一部分，我们发现有理域上定义为代数群的有限矩阵群G可以通过由域和G与Galois群的半直积构成的扭群环上的模来研究。由于可以为那些扭曲的群环定义特征标这一事实，仍然需要研究经典特征标论的哪些部分可以被改编。舒尔指数理论，特别是布劳尔-维特定理，将是人们感兴趣的中心。在项目的第二部分，计算有限表示群到所有L2型有限群上的满同态的有效方法已经推广到L3型和U3型。由此得到的理论见解将被发展成一般的特征标理论，例如可以应用于确定有限域上矩阵群的Aschbacher类。所发展的方法将用于研究代数整数环上的有限群表示和有限生成矩阵群。例如，Lubotzky关于SL(n，Z)的子群的结果是建设性的。

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.