Corollary 6.7

The subgroups of $\mathbb{Z}$ under addition are precisely the groups $n \mathbb{Z}$ under addition for $n \in \mathbb{Z}$.
Let $H$ be a subgroup of $\mathbb{Z}$. We already know that $\mathbb{Z}$ is a cyclic group with $1$ and $-1$ being its generators. Then by Theorem 6.6, we know that $H$ is cyclic as well. Now let $n \in \mathbb{Z}$ be the generator for $H$. Then $H=n\mathbb{Z}.\;\;\blacksquare$