Showing that $Aut~Z_p \simeq Z_{p-1}$ Okay. This is similar to a question I asked a few days ago. I am trying to show that $Aut~ Z_p$ is isomorphic to $Z_{p-1}$, where $Z_p$ denotes the congruence class of integers $\mod p$ and $p$ a prime. I have shown $Aut~ Z_p$ consists of $p-1$ elements, using the fact that a homomorphism is uniquely determined by how it maps $[1]_p$, the generator of $Z_p$; the only element it cannot be mapped to is $[0]_p$, otherwise $[1]_p \rightarrow [k]_p \neq [0]_p$ extends to an automorphism.

Now I am trying to show that $Aut~ Z_p$ is a cyclic, and show that mapping the generator of $Aut~ Z_p$ to $Z_{p-1}$ defines an isomorphism. However, I am having trouble showing this. I could use some hints.

Note: At this point, I don't know anything about fields (let alone rings), the order of an element, and related concepts. 

Edit: I noticed that the question was asked here, but from my cursory reading I didn't see any ideas I could use yet.
Hint:

Let $\varphi$ the automorphism of $\mathbf Z_p$ mapping $[1]$ to $[r]$. What does $\varphi^k$ map $[1]$ to? How can you use Lil' Fermat?
Hint:

Let $\varphi$ the automorphism of $\mathbf Z_p$ mapping $[1]$ to $[r]$. What does $\varphi^k$ map $[1]$ to? How can you use Lil' Fermat?