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

Let R be the ring of integers in a number field F, Lambda any R-order in a semi-simple F-algebra Sigma, Gamma any maximal R-order containing Lambda. We show in this paper that for all n greater than or equal to 2 rank K-n(Lambda) = rank G(n)(Lambda) = rank K-n(Gamma) = rank K-n(Sigma). Hence if G is a finite group, rank K-n(RG) = rank G(n)(RG)= rank K-n(FG). (C) 1999 Elsevier Science B.V. All rights reserved.