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

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