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

\(\ds x \circ y\)

\(=\)

\(\ds y \circ x\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {x \circ y} \circ x^{-1}\)

\(=\)

\(\ds \paren {y \circ x} \circ x^{-1}\)

Cancellation Laws

\(\ds \leadstoandfrom \ \ \)

\(\ds x \circ y \circ x^{-1}\)

\(=\)

\(\ds y \circ \paren {x \circ x^{-1} }\)

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

$\blacksquare$

Sources

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