Order of Group Element equals Order of Inverse

Theorem
Let $G$ be a group whose identity is $e$.

Then:
 * $\forall x \in G: \order x = \order {x^{-1} }$

where $\order x$ denotes the order of $x$.

Proof
By Powers of Group Elements: Negative Index:
 * $\paren {x^k}^{-1} = x^{-k} = \paren {x^{-1} }^k$

Hence:

Similarly:

A similar argument shows that if $x$ is of infinite order, then so must $x^{-1}$ be.

Hence the result.