Order of Group Element equals Order of Inverse

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

Then:
 * $\forall x \in G: \left|{x}\right| = \left|{x^{-1}}\right|$

where $\left|{x}\right|$ denotes the order of $x$.

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

Similarly:

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

Hence the result.