Commutation Property in Group

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Theorem

Let $\left({G, \circ}\right)$ be a group.

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x \circ y$	$=$	$\displaystyle y \circ x$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \left({x \circ y}\right) \circ x^{-1}$	$=$	$\displaystyle \left({y \circ x}\right) \circ x^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Cancellation Laws
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x \circ y \circ x^{-1}$	$=$	$\displaystyle y \circ \left({x \circ x^{-1} }\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Associativity
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x \circ y \circ x^{-1}$	$=$	$\displaystyle y \circ e$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Behaviour of Inverse
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x \circ y \circ x^{-1}$	$=$	$\displaystyle y$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Behaviour of Identity

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x \circ y\)	\(=\)	\(\displaystyle y \circ x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \left({x \circ y}\right) \circ x^{-1}\)	\(=\)	\(\displaystyle \left({y \circ x}\right) \circ x^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Cancellation Laws
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \circ y \circ x^{-1}\)	\(=\)	\(\displaystyle y \circ \left({x \circ x^{-1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \circ y \circ x^{-1}\)	\(=\)	\(\displaystyle y \circ e\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Behaviour of Inverse
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \circ y \circ x^{-1}\)	\(=\)	\(\displaystyle y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Behaviour of Identity