Example of a Short Exact Sequence I know that $A$ is a $\mathbb{Z}$-module. And I have a short exact sequence of the form 
$0 \rightarrow \mathbb{Z}/2\mathbb{Z} \rightarrow A \rightarrow \mathbb{Z}/2\mathbb{Z} \rightarrow 0$
is there anything to be said about $A$ ? That is to say, Can I find $A$?
I recall that $\mathbb{Z}/2\mathbb{Z}$ is neither projective nor injective as a $\mathbb{Z}$-module so the sequence probably doesn't split. Or does it?
 A: A short exact sequence always could split. I mean that $0\to M\to M\oplus N\to N\to 0$ is always exact when the maps are the inclusion of $M$ as the first coordinate and the projection onto $N$. But you're right that you can't guarantee this splits via the theorems you alluded to. So the point of a question like this is to see whether $A$ could be anything other than $\mathbb{Z}/2\oplus\mathbb{Z}/2$.
So, we know $\mathbb{Z}/2$ injects into $A$, i.e. there's a two-element subgroup $B\subset A$. Since $A/B\simeq \mathbb{Z}/2$ as well, $B$ is a two-element subgroup of index 2. That tells us $|A|=4,$ so $A$ could only be $\mathbb{Z}/2\oplus\mathbb{Z}/2$ or $\mathbb{Z}/4$. And I'll leave it to you to check whether it could be the latter.
A: Well, $\Bbb Z$-module means abelian group. $A$ needs to have $4$ elements. So either $A=\Bbb Z/4\Bbb Z$ or $A=\Bbb Z/2\Bbb Z\times\Bbb Z/2\Bbb Z$.
Can you fit either one in such an exact sequence?
A: I'm no expert in homology theory but I am aware of the fact that you can actually classify such exact sequences by $\mathrm{Ext}$ groups. Of course, in this case it would be rather foolish thing to use such apparatus in fairly simple case, however, in general setting it might be usefull to check this.
I recommend to look up $\mathrm{Ext}$ in Dummit & Foote Abstract Algebra book
Hope this helps.
