Ranks of Kn and Gn of orders and group rings of finite groups over integers in number fields

Abstract

Let R be the ring of integers in a number field F, Λ any R-order in a semi-simple F-algebra σ, Γ any maximal R-order containing Λ. We show in this paper that for all n ≥ 2 rank K n ( Λ) = rank G n ( Λ) = rank K n ( Γ) = rank K n ( σ). Hence if G is a finite group, rank K n ( RG) = rank G n ( RG) = rank K n ( FG).