Product of Finite Abelian Group I need to proof for a finite abelian group $G$ that for all $x\in G$ we have $\prod_{g\in G}xg=\prod_{g\in G}g$.
I figured that using the commutative property $\prod_{g\in G}xg=x^n\prod_{g\in G}g$. Which would leave us to proof $x^n=e$, where $e$ is the identity element.
Hopefully I'm just missing something obvious here, because I feel like smashing my head against the table now.

The map $g\mapsto xg$ is a permutation of $G$. hence $\prod xg$ is the same as $\prod g$, just in a different order. As $G$ is abelian, different order doesn't matter.
