---
tag: 0PMT9YJZVW
kind: statement
role: proposition
name: Order of an inverse
msc: [20A05]
uses: [0JPYSD5E0B, 00DRK90Z6K]
sources:
  - url: https://en.wikipedia.org/wiki/Order_(group_theory)
    locator: Order and structure
---

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
Then $g$ and its inverse $g^{-1}$ have the same [[order|0WE2NHZJ7B]].
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 $4$.