Commutation of Inverses in Monoid

Theorem
Let $$\left({S, \circ}\right)$$ be a monoid whose identity is $$e_S$$. Let $$x, y \in S$$ such that $$x$$ and $$y$$ are both invertible.

Then $$x$$ commutes with $$y$$ iff $$x^{-1}$$ commutes with $$y^{-1}$$.

Necessary Condition
Let $$x$$ commute with $$y$$. Then:

$$ $$ $$

So $$x^{-1}$$ commutes with $$y^{-1}$$.

Sufficient Condition
Now let $$x^{-1}$$ commute with $$y^{-1}$$.

From the above, $$\left({x^{-1}}\right)^{-1}$$ commutes with $$\left({y^{-1}}\right)^{-1}$$.

From Inverse of an Inverse, $$\left({x^{-1}}\right)^{-1} = x$$ and $$\left({y^{-1}}\right)^{-1} = y$$.

Thus $$x$$ commutes with $$y$$.