# 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 |

$\blacksquare$

## Sources

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