Order of an inverse

Statement

Let $G$ be a group and let $g \in G$. Then $g$ and its inverse $g^{-1}$ have the same order. In particular, $g$ has infinite order if and only if $g^{-1}$ has infinite order.

Remarks

For each positive integer $n$, $(g^{-1})^n = (g^n)^{-1}$, and $g^n = e$ if and only if $(g^n)^{-1} = e$. For example, in $\{1, i, -1, -i\}$, both $i$ and $i^{-1} = -i$ have order $4$.

Explanations

L1 human

The inverse of an element walks the same path back to the identity, in the opposite direction. It needs the same number of steps. A rotation by $90$ degrees and a rotation by $-90$ degrees both have order $4$.