A nontrivial finite abelian group is cyclic if and only if it has one invariant factor

Given: The objects and hypotheses in the statement.

For every finite abelian group $G$ there is a unique list $1<n_1\mid\cdots\mid n_r$ such that $G\cong C_{n_1}\times\cdots\times C_{n_r}$. Moreover $|G|=n_1\cdots n_r$. The trivial group corresponds to the empty list and empty product. (Fundamental theorem of finite abelian groups: invariant-factor form).

If $G\cong C_{n_1}\times\cdots\times C_{n_r}$ with $1<n_1\mid\cdots\mid n_r$, then $|G|=n_1\cdots n_r$ and $\exp(G)=n_r$. For the empty list, $|G|=\exp(G)=1$. (Invariant factors determine the order and exponent of a finite abelian group).

If $G=\langle g\rangle$ is cyclic, then exactly one of the following applies: - if $g$ has infinite order, $G\cong(\mathbb Z,+)$; - if $g$ has finite order $n$, necessarily $n\ge1$, then $G\cong(\mathbb Z/n,+)$. (Every cyclic group is isomorphic to $(\mathbb Z,+)$ or to $(\mathbb Z/n,+)$ for its finite order $n\ge1$).