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 $x=y$)?

Hint: What is that relation? In fact, $$x\sim y~~~\text{iff}~~~\exists g\in G,~ x=gyg^{-1}$$ So what would happen if we knew for all $x,y\in G, xy=yx$ or for all $$x,y\in G, x=yxy^{-1}$$
