Group Element is Self-Inverse iff Order 2

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Theorem

Let $\struct {S, \circ}$ be a group whose identity is $e$.


An element $x \in \struct {S, \circ}$ is self-inverse if and only if:

$\order x = 2$


Proof

Let $x \in G: x \ne e$.

\(\displaystyle \order x\) \(=\) \(\displaystyle 2\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x \circ x\) \(=\) \(\displaystyle e\) Definition of Order of Group Element
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle x^{-1}\) Equivalence of Definitions of Self-Inverse

$\blacksquare$


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