13.1 (1995)

Let $U(K)$ denote the group of units of a ring $K$. Let $G$ be a finite group, $\mathbb{Z}G$ the integral group ring of $G$, and $\mathbb{Z}_p G$ the group ring of $G$ over the residues modulo a prime number $p$. Describe the homomorphism from $U(\mathbb{Z}G)$ into $U(\mathbb{Z}_p G)$ induced by reducing the coefficients modulo $p$. More precisely, find the kernel and the image of this homomorphism and an explicit transversal over the kernel.