2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141442Let $l$ be an odd prime number and $H$ a finite abelian $l$-group. We determine the unit group of $Λ_\wedge[H]$ (the completion of the localization at $l$ of $\Bbb{Z}_l[[T]][H]$) as well as the kernel and cokernel of the integral logarithm $L:Λ_\wedge[H]^\times\to Λ_\wedge[H]$, which appears in non-commutative Iwasawa theory.Number Theory11R23text