Möbius Inversion Formula/Abelian Group

Theorem

Let $G$ be an abelian group.

Let $f, g: \N \to G$ be mappings.

Then

$\ds \map f n = \prod_{d \mathop \divides n} \map g d$

if and only if:

$\ds \map g n = \prod_{d \mathop \divides n} \map f d^{\mu \paren {\frac n d} }$

where:

$d \divides n$ denotes that $d$ is a divisor of $n$

$\mu$ is the Möbius function.

Proof

This theorem requires a proof. In particular: The proof should go in the same way as in the main theorem ($G = \R$ or $\C$). You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.