Prove that if $G$ is an abelian group then the conjugation equivalence relation is the identity relation ($x\sim y$ if and only if $x=y$). How do you prove that if the group $G$ is abelian, then the conjugation equivalence relation is the identity relation ($x \sim y$ iff $x=y$)?

If you are in an abelian group then conjugation does nothing since $ghg^{-1} = h$. So under the equivalence relation, $h$ can only be equivalent to itself.
