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 $x \in G: x \ne e$.
\(\ds \order x\) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x \circ x\) | \(=\) | \(\ds e\) | Definition of Order of Group Element | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds x^{-1}\) | Equivalence of Definitions of Self-Inverse |
$\blacksquare$
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Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $9$