TITLE: For what Abelian Group $A$ is there the following exact sequence?
QUESTION [6 upvotes]: The question is:

For what kind of Abelian group $A$ is there a short exact sequence:
  $$
0\to\mathbb{Z}\to A\to\mathbb{Z}_{n}\to0.
$$

This is an exercise from Allen Hatcher's Algebraic Topology textbook. I could name a few $A$s like $\mathbb{Z}, \mathbb{Z}\oplus\mathbb{Z}_{n},$ etc.
But I have no idea what kind of general $A$ can but put in this short exact sequence. Does anyone know if there is some topological background for this problem?
My result is:
$A=Z\oplus Z_{d}$, with some d|n.
My method is to consider a commutative diagram where the upper sequence is:
$0 \to Z \to Z \oplus Z \to Z \to 0 $
and the bottom is the exact sequence in the problem. We could try to construct (using the exactness of the bottom line) a surjective map from $Z\oplus Z$ to A to make the diagram commutative.

REPLY [2 votes]: The point of the question is that there aren't that many $A$ that fit in such a sequence.  A useful lemma to prove might be the following:

Let $0 \to X \to Y \to Z\to 0$ is a short exact sequence of Abelian groups, and $X$ and $Z$ are generated by $m$ and $n$ elements, respectively.  Then $Y$ has a generating set with $m+n$ or fewer elements.

In your example, this would imply that $A$ is an Abelian group with at most $2$ generators.  Such groups are classified by the structure theorem, and it only remains to see which of those fit into such an exact sequence.