[[Group order]]
# The order of an element and its inverse are the same

Given a group $G$ and an element $a \in G$ with order $\abs{a} = n$,
the order of the inverse $\abs{a^{-1}} = n$. #m/thm/group

> [!check]- Proof
> By the uniqueness of the inverse, $a^n = e$ iff $a^{-n} = e$.
> Therefore the orders of $a$ and $a^{-1}$ must be equal,
> since neither can have a lower order than the other.
> <span class="QED"/>


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