---
tag: 08V7JHS3QZ
kind: statement
role: corollary
name: Order of an element divides the order of the group
msc: [20D60]
uses: [3C53TD, 031M8HX7DA]
sources:
  - url: https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory)
    locator: Applications
---

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $g \in G$.
Then the [[order|0WE2NHZJ7B]] of $g$ divides the [[order|0B85S2V2DR]] $|G|$ of $G$.

## Remarks

The order of $g$ is the order of the cyclic subgroup $\langle g \rangle$, [[by this fact|031M8HX7DA]], and [[Lagrange's theorem|3C53TD]] applies to $\langle g \rangle$.
For example, in a group of order $10$, each element has order $1, 2, 5$ or $10$.

## Explanations

### L1 human

An element generates a subgroup with as many elements as its order.
The size of a subgroup always divides the size of the group.
So the order of an element divides the order of the group.