How to show that $Z_{p^2}\ncong Z_p\rtimes Z_p$, where $p$ is a prime? My thoughts:
1.First show that $Z_{p^2}\ncong Z_p\times Z_p$.
2.Recall that group of order $p^2$ is isomorphic to either $Z_{p^2}$ or $Z_p\times Z_p$, so show that $Z_p\rtimes Z_p\cong Z_p\times Z_p$. 
But I am stuck on the second step, how should I continue? And is there a better way to solve this?

To talk about a semi-direct product, you have to describe how one group
acts on the other.  Why don't you begin by computing the possible ways
that $\mathbb Z_p$ can act as automorphisms of itself?  Then you might have a better shot of proving your step 2.
