We study groups of matrices SGL(n)(Z Gamma) of augmentation one over t
he 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 th
e 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 Ac
ademic Press, Inc.