Exercise 1.2.14 (Set of generators for $\mathbb{Z}$)

Proof. The singleton $\{1\}$ is a set of generators of $\mathbb{Z}$: any subgroup $H$ of $\mathbb{Z}$ which contains $1$ contains also $n\cdot 1=n$ for every $n\in \mathbb{Z}$, so $H=\mathbb{Z}$. This shows that

(Other sets of generators are $\{-1\}$, or any subset of $\mathbb{Z}$ which contains $1$ (or $-1$)). □