---
tag: 0VVPQNVM8S
kind: statement
role: proposition
name: Order of a power
msc: [20A05]
uses: [0JPYSD5E0B, 0F5QC101Q7]
sources:
  - url: https://en.wikipedia.org/wiki/Order_(group_theory)
    locator: Order and structure
---

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$ be an element of finite [[order|0WE2NHZJ7B]] $n$.
Let $k$ be an integer.
Then the order of $g^k$ is $n / \gcd(n, k)$.

## Remarks

Here $\gcd(n, 0) = n$, so that $g^0 = e$ has order $1$.
In particular, $g^k$ has order $n$ if and only if $\gcd(n, k) = 1$.
For example, if $g$ has order $12$, then $g^8$ has order $12 / 4 = 3$.

## Explanations

### L1 human

A power of an element returns to the identity faster than the element itself, or at the same time.
If $g$ has order $12$, then $g^8$ is at the identity after $k$ steps exactly when $12$ divides $8k$, that is, when $3$ divides $k$.
Thus $g^8$ has order $3$.