Prove that if $G$ is an abelian group then the conjugation equivalence relation is the identity relation ($x\sim y$ if and only if $x=y$).

Question

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

Answer 1

We let $x,y\in G$. Given that $x\sim y$ is an equivalence relation then we need to show that, $x y = y x$ is abelian then it follows that $$x=x y y^{-1}= x[y y^{-1}]=(xy)y^{-1}=(y x)x^{-1}=ye=y.$$

Answer 2

If I understood the question correctly assume that ~ is the conjugation relation, then we have x and y are related if and only if $$x=z^{-1}yz=z^{-1}zy=y$$ since the group is abelian.

Answer 3

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 $$x,y\in G, x=yxy^{-1}$$

Answer 4

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.