Commutator is Identity iff Elements Commute

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Theorem

Let $G$ be a group whose identity is $e$.

Let $x, y \in G$.

Let $\sqbrk {x, y}$ denote the commutator of $x$ and $y$.


Then $\sqbrk {x, y} = e$ if and only if $x$ and $y$ commute.


Proof

As $G$ is a group, it is by definition a monoid.


Hence Product of Commuting Elements with Inverses applies:

$x y x^{-1} y^{-1} = e = x^{-1} y^{-1} x y$

$\blacksquare$


Sources