Commutation with Inverse

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Theorem

Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$.

Let $x, y \in S$ such that $y$ is invertible.

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

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle y^{-1} \circ x$	$=$	$\displaystyle y^{-1} \circ \left({x \circ \left({y \circ y^{-1} }\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Invertibility of $y$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle y^{-1} \circ \left({\left({x \circ y}\right) \circ y^{-1} }\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Associativity of $\circ$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle y^{-1} \circ \left({\left({y \circ x}\right) \circ y^{-1} }\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Commutativity of $x$ and $y$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({y^{-1} \circ \left({y \circ x}\right)}\right) \circ y^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Associativity of $\circ$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({\left({y^{-1} \circ y}\right) \circ x}\right) \circ y^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Associativity of $\circ$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle x \circ y^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Invertibility of $y$

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

$\Box$

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

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

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

Thus $x$ commutes with $y$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle y^{-1} \circ x\)	\(=\)	\(\displaystyle y^{-1} \circ \left({x \circ \left({y \circ y^{-1} }\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Invertibility of $y$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle y^{-1} \circ \left({\left({x \circ y}\right) \circ y^{-1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity of $\circ$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle y^{-1} \circ \left({\left({y \circ x}\right) \circ y^{-1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Commutativity of $x$ and $y$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({y^{-1} \circ \left({y \circ x}\right)}\right) \circ y^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity of $\circ$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\left({y^{-1} \circ y}\right) \circ x}\right) \circ y^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity of $\circ$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x \circ y^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Invertibility of $y$