Writing $\mathbb{Z}_{p^2}$ as central extension of $\mathbb{Z}_{p}$ by $\mathbb{Z}_{p}$

Question

I would like to prove that $\mathbb{Z}_{p^2}$, where $p$ is prime, can be constructed as a central extension of $\mathbb{Z}_{p}$ by $\mathbb{Z}_{p}$.
So I need $$0 \rightarrow \mathbb{Z_p} \rightarrow \mathbb{Z}_{p^2} \rightarrow \mathbb{Z}_{p} \rightarrow 0$$ I am not sure how to show that this sequence is short exact.
Also, can $\mathbb{Z}_{p^2}$ be written as a semidirect product?

Answer 1

I'm going to label your maps. We'd like to find $f$ and $g$ that turn the sequence below into a central extension: $$0 \rightarrow \mathbb{Z}_p \xrightarrow{f} \mathbb{Z}_{p^2} \xrightarrow{g} \mathbb{Z} \rightarrow 0$$ Since $\mathbb{Z}_{p^2}$ is abelian, we know that $Im(f)\subset Z(\mathbb{Z}_{p^2})$ for any group homomorphism $f$. So really, to show that it's a central extension, we just need to find $f$ and $g$ so that the sequence is exact.

I don't want to give the answer away, but note that $\mathbb{Z}_{p^2}$ contains a subgroup isomorphic to $\mathbb{Z}_p$. Can you see what this subgroup would be? Once you've found it, again note that the ambient group is abelian, so all subgroups are normal. This might cause you to think about quotient maps.

See Arturo's comment for the question about whether this can be realized as a semidirect product.