Chapter 3.3, Problem 42E

Reword Definition $3.17$ for an additive group $G$.

Definition $3.17$: Let $G$ be a group. For any $a\in G$, the subgroup

$H=\left\{x\in G|x=a^{n}forn\in \mathbb{Z}\right\}$ is the subgroup generated by $a$ and is denoted by $\langle a\rangle$. The element $a$ is called a generator of $H$. A given subgroup $K$ of $G$ is a cyclic subgroup if there exists an element $b$ in $G$ such that

$K=\langle b\rangle =\left\{y\in G|y=b^{n}forsomen\in \mathbb{Z}\right\}$.

In particular, $G$ is a cyclic group if there is an element $a\in G$ such that $G=\langle a\rangle$.