---
tag: 8WJPSS7TE9
kind: example
name: The even integers form a subgroup of the integers
msc: [20A05]
instance_of: [0X1JXT47GA, 0CB0DX5BVV]
uses: [0C4JS9RF5M, 0N0PXCTYA7, 0W77CVBCHQ]
sources:
  - url: https://en.wikipedia.org/wiki/Subgroup
    locator: Other examples
---

## Example

Let $G = (\mathbb{Z}, +)$ be the [[integers|0C4JS9RF5M]] with addition, and let $2\mathbb{Z}$ be the set of even integers.
The identity $0$ is even.
The sum of two even integers is even, and the negative of an even integer is even.
Thus $2\mathbb{Z}$ is a [[subgroup|0N0PXCTYA7]] of $G$.
$G$ is abelian, so $g + 2\mathbb{Z} = 2\mathbb{Z} + g$ for each $g$. Thus $2\mathbb{Z}$ is [[normal|0W77CVBCHQ]] in $G$.

## Remarks

In the same way, $n\mathbb{Z}$ is a subgroup of $\mathbb{Z}$ for each $n \ge 0$.