Group Element is Self-Inverse iff Order 2

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

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

$\blacksquare$


This article is complete as far as it goes, but it could do with expansion.
In particular: Free generalisation to monoids
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Sources

  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $9$
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