I'm doing some exercises in Hatcher:
Determine whether there is a short exact sequence $ 0 \rightarrow \mathbb{Z}_4 \xrightarrow{f} \mathbb{Z}_8 \oplus \mathbb{Z}_2 \xrightarrow{g} \mathbb{Z}_4 \rightarrow 0$. More generally, which abelian groups $A$ fit into a short exact sequence $0 \rightarrow \mathbb{Z}_{p^m} \xrightarrow{f} A \xrightarrow{g} \mathbb{Z}_{p^n} \rightarrow 0 $ where $p$ prime. What about 0 \rightarrow \mathbb{Z} \xrightarrow{f} A \xrightarrow{g} \mathbb{Z}_n \rightarrow 0.
Can you tell me if this is right:
In the first case, because $f$ injective, it has to map the generator, $1$, to an element of order $4$. There are $2$ essentially different elements of order $4$: $(2,1)$ and $(2,0)$. In both cases I get a contradiction if they are in $ker g$, so there cannot be such an exact sequence.
In the next case, $\mathbb{Z}_{p^m}$ is cyclic, let's say it's generated by $c$. Then $f(c)$ has order $p^m$. So $ker g$ has order $p^m$. Therefore $A$ has to have at least order $p^{n+m}$. Now I'm not sure how to proceed.
Can I deduce anything else about $A$?
Many thanks for your help!