Units of twisted group rings and their correlations to classical group rings

Abstract

Given a central extension Γ of some normal subgroup N by a group G, we study the group ring RΓ over some domain R. We obtain a direct sum decomposition in terms of various twisted group rings of G and concrete information on the kernel and cokernel of the projections. This allows to kick-start the investigation of the unit group U (RΓ) via the unit group of twisted group rings. Among others we construct a new generic family of units therein. As an application hereof we are able to obtain a complete description of the unit group of the integral group ring of G × C n 2 from data of G.Contents 1. Introduction 2. Cohomological preliminaries 3. On the decomposition of the twisted group algebras of general extensions and transgression 3.1. Concrete decomposition 3.2. Transgression and natural morphisms as projections 3.3. The case of an α-representation group of G 4. When is the unit group of a twisted group ring finite? 5. Correlations between R[Γ] and R α [G] -a unit group point of view 5.1. On the cokernel of the generalized transgression map 5.2. On the kernel of the generalized transgression map 6. Twisting bicyclic units and their contribution 6.1. Generalized byclic units versus elementary matrices 6.2. Twisted bicyclic units -general theory 6.3. On twisted bicyclic units in cokernel of the transgression map 7. Method to describe Udescription for elementary abelian 2-groups 8.1. The starting case C 2 × C 2 8.2. The general case 9. Description of U(Z[D 8 × C n 2 ]) References