# Group Element Commutes with Inverse

## Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $x \in G$.

Then:

$x \circ x^{-1} = x^{-1} \circ x$

That is, $x$ commutes with its inverse $x^{-1}$.

## Proof

By definition of inverse element:

$x \circ x^{-1} = e = x^{-1} \circ x$

Hence the result by definition.