Subgroups of the integers

Statement

Let $H$ be a subgroup of the additive group of integers $(\mathbb{Z}, +)$. 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 $\{0\}$.