  Let $l$ be an odd prime number and $H$ a finite abelian $l$-group. We
determine the unit group of $\Lambda_\wedge[H]$ (the completion of the
localization at $l$ of $\Bbb{Z}_l[[T]][H]$) as well as the kernel and cokernel
of the integral logarithm $L:\Lambda_\wedge[H]^\times\to \Lambda_\wedge[H]$,
which appears in non-commutative Iwasawa theory.
