---
tag: 0SRATCFF1W
kind: statement
role: proposition
name: Subgroups of the integers
msc: [20K27]
uses: [0KB1ND7XDS, 0BSCJA5G53]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Subgroups
---

## Statement

Let $H$ be a [[subgroup|0X1JXT47GA]] of the [[additive group of integers|0BSCJA5G53]] $(\mathbb{Z}, +)$.
Then there is exactly one integer $n \geq 0$ with $H = n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}$.

## Remarks

When $H \neq \{0\}$, the integer $n$ is the smallest positive element of $H$.
For $m, n \geq 0$, $m\mathbb{Z} \subseteq n\mathbb{Z}$ holds exactly when $n$ divides $m$.
A consequence is that, for integers $a$ and $b$, the subgroup $a\mathbb{Z} + b\mathbb{Z}$ is $d\mathbb{Z}$ with $d = \gcd(a, b)$, which gives Bezout's identity.

## Explanations

### L1 human

Each subgroup of the integers is the set of multiples of one number.
For example, the set of all sums $4x + 6y$ with integers $x, y$ is the set of multiples of $2$.
The number $0$ gives the trivial subgroup $\{0\}$.