Möbius Inversion Formula/Abelian Group

Theorem
Let $G$ be an abelian group.

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

Then


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




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

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