Commutation of Inverses in Monoid

Theorem
Let $\struct {S, \circ}$ 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 $y$ $x^{-1}$ 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, $\paren {x^{-1} }^{-1}$ commutes with $\paren {y^{-1} }^{-1}$.

From Inverse of Inverse in Monoid, $\paren {x^{-1} }^{-1} = x$ and $\paren {y^{-1} }^{-1} = y$.

Thus $x$ commutes with $y$.