---
tag: 0F5QC101Q7
kind: statement
role: proposition
name: g to the n equals 1 iff the order divides n
aliases: [power-is-identity-iff-order-divides]
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]] with identity element $e$, and let $g \in G$ be an element of finite [[order|0WE2NHZJ7B]] $m$.
Let $n$ be an integer.
Then $g^n = e$ if and only if $m$ divides $n$.

## Remarks

The proof divides $n$ by $m$ with remainder: $n = qm + r$ with $0 \leq r < m$, so that $g^n = (g^m)^q g^r = g^r$.
By the minimality of $m$, the power $g^r$ is $e$ only when $r = 0$.
A consequence is that $g^a = g^b$ if and only if $a \equiv b \pmod m$.

## Explanations

### L1 human

An element of order $m$ comes back to the identity exactly after $m$, $2m$, $3m$ steps, and so on.
It is never at the identity after a number of steps that is not a multiple of $m$.
For example, a rotation by $90$ degrees is the identity after $8$ steps, but not after $6$ steps.