---
tag: 0SEX9F9J5Z
kind: definition
name: Order of a group
msc: [20A05]
defines: 0B85S2V2DR
uses: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Order_(group_theory)
    locator: lead section
---

## Definition

Let $G$ be a [[group|G5S1M9]].
The order of $G$, written $|G|$, is the cardinality of the underlying set of $G$.
When $G$ is finite, $|G|$ is the number of elements of $G$.
When $G$ is infinite, $G$ has infinite order.

## Remarks

The [[trivial group|0BNPC9850Y]] has order $1$, and the [[group of $n$th roots of unity|0WX11EA3F3]] has order $n$.
The group $(\mathbb{Z}, +)$ has infinite order.
[[Lagrange's theorem|3C53TD]] says that the order of a subgroup of a finite group divides the order of the group.

## Explanations

### L1 human

The order of a group tells you how big the group is.
For example, the rotations of a square that map the square to itself form a group of order $4$.
The word "order" has a second meaning in group theory: the order of an element.