Finite matrix groups over nilpotent group rings

We study groups of matrices SGL(n)(Z Gamma) of augmentation one over the integral group ring Z(G)amma of a nilpotent group Gamma. We relate the torsion of SGL(n)(Z Gamma) to the torsion of Gamma. We prove that all abelian p-subgroups of SGL(n)(Z Gamma) can be stably diagonalized. Also, all finite subgroups of SG(n),(Z Gamma) can be embedded into the diagonal Gamma(n) < SGL(n)(Z Gamma). We apply matrix results to show that if Gamma is nilpotent-by-(II'-finite) then all finite II-groups of normalized units in Z Gamma can be embedded into Gamma. (C) 1996 Academic Press, Inc.