Properties of Inverses of Commuting Elements
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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.