G is a group and $x,y\in G$.
Prove that $xy=yx$ iff $y^{-1}xy=x$ iff $x^{-1}y^{-1}xy=1$.
If we have to show that we have i) iff ii) iff iii), then just prove that $i)\Rightarrow ii)$, $ii)\Rightarrow iii)$ and $iii)\Rightarrow i)$.