Möbius Inversion Formula/Abelian Group

Theorem
Let $G$ be an abelian group.

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

Then


 * $\displaystyle f \left({n}\right) = \prod_{d \mathop \backslash n} g \left({d}\right)$




 * $\displaystyle g \left({n}\right) = \prod_{d \mathop \backslash n} f \left({d}\right) ^{\mu \left({\frac n d}\right)}$

where:
 * $d \mathrel \backslash n$ denotes that $d$ is a divisor of $n$
 * $\mu$ is the Möbius function.