# Properties of Inverses of Commuting Elements

## Theorems

Let $\struct {S, \circ}$ be a monoid with identity element $e_S$.

### Commutation with Inverse in Monoid

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

Then $x$ commutes with $y$ if and only if $x$ commutes with $y^{-1}$.

### Commutation of Inverses in Monoid

Let $x, y \in S$ such that $x$ and $y$ are both invertible.

Then $x$ commutes with $y$ if and only if $x^{-1}$ commutes with $y^{-1}$.

### Inverse of Commuting Pair

Let $x, y \in S$ such that $x$ and $y$ are both invertible.

Then $x$ commutes with $y$ if and only if:

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

### Conjugate of Commuting Elements

Let $x, y \in S$ such that $x$ and $y$ are both invertible.

Then $x \circ y \circ x^{-1} = y$ if and only if $x$ and $y$ commute.

### Product of Commuting Elements with Inverses

Let $x, y \in S$ such that $x$ and $y$ are both invertible.

Then:

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

if and only if $x$ and $y$ commute.