The integral logarithm in Iwasawa theory: an exercise

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Let l be an odd prime number and H a finite abelian l-group. We determine the unit group of Λ[H] (the completion of the localization at l of Zₗ[[T]][H]) as well as the kernel and cokernel of the integral logarithm L:Λ[H]×→ Λ[H], which appears in non-commutative Iwasawa theory.