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

 $\displaystyle \order x$ $=$ $\displaystyle 2$ $\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

$\blacksquare$