## Two group elements commute if and only if they fix each other under conjugation if and only if their commutator is trivial

Let $G$ be a group, and let $x,y \in G$. Prove that $xy = yx$ if and only if $y^{-1}xy = x$ if and only if $x^{-1}y^{-1}xy = 1$.

The first “if and only if” follows by left multiplying by $y$ $(\Leftarrow)$ or $y^{-1}$ $(\Rightarrow)$. The second follows by left multiplying by $x$ $(\Leftarrow)$ or $x^{-1}$ $(\Rightarrow)$. $\blacksquare$