---
tag: 77N4TVRM15
kind: example
name: Orders of the elements of the integers modulo 6
msc: [20A05]
instance_of: [0WE2NHZJ7B]
illustrates: [08V7JHS3QZ]
uses: [03VFZ2WETT, 0JPYSD5E0B]
sources:
  - url: https://en.wikipedia.org/wiki/Order_(group_theory)
    locator: Order and structure
---

## Example

Let $\mathbb{Z}/6\mathbb{Z} = \{0, 1, 2, 3, 4, 5\}$ be the [[integers modulo $6$|03VFZ2WETT]] with addition.
The element $2$ has [[order|0JPYSD5E0B]] $3$, because $2 + 2 + 2 = 6 = 0$ and $2 + 2 = 4 \ne 0$.
The element $3$ has order $2$, and the elements $1$ and $5$ have order $6$. The element $4$ has order $3$.
Each of the orders $1$, $2$, $3$ and $6$ divides $6$, as the [[result on element orders|08V7JHS3QZ]] states.

## Remarks

The group is abelian, but not each element has order $6$.