3 added 1 characters in body

In the finitely generated case it follows by the structure theorem that the unique possibilities are $G=\mathbb{Z}$ and $G=\mathbb{Z}/({p_1}^{e_1})\oplus \mathbb{Z}/({p_2}^{e_2})\oplus\cdots\oplus \mathbb{Z}/({p_k}^{e_k})$, where $p_1,\dots,p_k$ are distinct primes.

Edit: in a word, the unique possibility is that $G$ is cyclic, as remarked by Jack Schmidt below.

2 added 104 characters in body

In the finitely generated case it follows by the structure theorem that the unique possibilities are $G=\mathbb{Z}$ and $G=\mathbb{Z}/({p_1}^{e_1})\oplus \mathbb{Z}/({p_2}^{e_2})\oplus\cdots\oplus \mathbb{Z}/({p_k}^{e_k})$, where $p_1,\dots,p_k$ are distinct primes.

Edit: in a word the unique possibility is that $G$ is cyclic, as remarked by Jack Schmidt below.

1

In the finitely generated case it follows by the structure theorem that the unique possibilities are $G=\mathbb{Z}$ and $G=\mathbb{Z}/({p_1}^{e_1})\oplus \mathbb{Z}/({p_2}^{e_2})\oplus\cdots\oplus \mathbb{Z}/({p_k}^{e_k})$, where $p_1,\dots,p_k$ are distinct primes.