# Commutation Property in Group

## 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

$\blacksquare$