Commutation Property in Group

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Theorem

Let $\struct {G, \circ}$ be a group.

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


Proof

\(\displaystyle x \circ y\) \(=\) \(\displaystyle y \circ x\)
\(\displaystyle \iff \ \ \) \(\displaystyle \paren {x \circ y} \circ x^{-1}\) \(=\) \(\displaystyle \paren {y \circ x} \circ x^{-1}\) Cancellation Laws
\(\displaystyle \iff \ \ \) \(\displaystyle x \circ y \circ x^{-1}\) \(=\) \(\displaystyle y \circ \paren {x \circ x^{-1} }\) Definition of Associative Operation
\(\displaystyle \iff \ \ \) \(\displaystyle x \circ y \circ x^{-1}\) \(=\) \(\displaystyle y \circ e\) Definition of Inverse Element
\(\displaystyle \iff \ \ \) \(\displaystyle x \circ y \circ x^{-1}\) \(=\) \(\displaystyle y\) Definition of Identity Element

$\blacksquare$


Sources