A Cancellation Theorem For Modules Over Integral Group Rings

A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups G for which the integral group ring DOUBLE-STRUCK CAPITAL ZG has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that DOUBLE-STRUCK CAPITAL ZG has SFC provided at most one copy of the quaternions occurs in the Wedderburn decomposition of the real group ring Double-struck capital RG. This generalises the Eichler condition in the case of integral group rings.