Algebraic group techniques for finite(ly presented) groups

• Applicant: Professor Dr. Wilhelm Plesken
• Subject Area: Mathematics
• Term: from 2009 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 123863751

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1388: Representation Theory