Abstract: A conjecture of Zassenhaus says that if G is a finite group then any unit u = ∑ u(g) g of finite order in ZG is conjugate in QG to ± g, for some g ∈ G. This is known to be equivalent to saying that ũ(g0) = ∑h ∼ g0, u(h) is nonzero for a unique conjugacy class Cg0 of elements of G. We prove that this latter condition holds for infinite nilpotent groups as well. We also study the possibility of stable diagonalization of torsion matrices over ZG.