Two cosets commute precisely when the subgroup contains the commutator of two representatives

Let G be a group and N \leq G a normal subgroup. Prove that x{}N and yN commute in G/N if and only if x^{-1}y^{-1}xy \in N. (The element [x,y] = x^{-1}y^{-1}xy is called the commutator of x and y in G.)


We have (xN)(yN) = (yN)(xN) if and only if (xy)N = (yx)N, if and only if (yx)^{-1}(xy) = x^{-1}y^{-1}xy \in N.

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