Induction theorems for Grothendieck groups and Whitehead groups of finite groups

Abstract

Abstract : For an abelian category C, the Grothendieck group K(O)(C) of C is an abelian group which solves the universal problem of finding 'additive' maps of Obj C into abelian groups. The Whitehead group K(1) C is the abelian group which furnishes a 'universal determinant theory' for C. The dissertation conducts a survey of certain techniques useful in the study of these groups. The groups are treated in some generality in the first chapter, then the survey is restricted to group rings, i.e. to the case when C is the category of Z pi-modules or else the category of projective Z pi-modules, for finite groups pi. In this setting, the crucial (and most subtle) question seems to be that of determining the torsion of the Whitehead group, and this is chiefly what is done in the last two chapters. (Author)