Commutation Property in Group

Theorem

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

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

$\ds x \circ y$

$=$

$\ds y \circ x$

$\ds \leadstoandfrom \ \ $

$\ds \paren {x \circ y} \circ x^{-1}$

$=$

$\ds \paren {y \circ x} \circ x^{-1}$

$\ds \leadstoandfrom \ \ $

$\ds x \circ y \circ x^{-1}$

$=$

$\ds y \circ \paren {x \circ x^{-1} }$

Definition of Associative Operation

$\ds \leadstoandfrom \ \ $

$\ds x \circ y \circ x^{-1}$

$=$

$\ds y \circ e$

Definition of Inverse Element

$\ds \leadstoandfrom \ \ $

$\ds x \circ y \circ x^{-1}$

$=$

$\ds y$

Definition of Identity Element

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $6 \ \text{(i)}$