let p be a prime. Show that $\displaystyle
Aut\left( {\mathbb{Z}_p } \right) \simeq \mathbb{Z}_{p - 1}
$
and show that $\displaystyle
Aut\left( {\mathbb{Z}_8 } \right) \simeq \mathbb{Z}_2
$
In general, $\displaystyle \mbox{Aut } \mathbb{Z}_n \cong \mathbb{Z}_n^\times$.
It's not trivial at all that $\displaystyle \mathbb{Z}_p^\times$ is cyclic. It's not very difficult to show once you know it, but you are probably not expected to come up with the proof. Instead you are probably expected to use this as a fact.
The second one is false - you have $\displaystyle \mbox{Aut } \mathbb{Z}_8 \cong \mathbb{Z}_8^\times \cong \mathbb{Z}_2 \times \mathbb{Z}_2 $.
Yes that is what I mean! Sorry for not being more clear.
To see that $\displaystyle
\mbox{Aut } \mathbb{Z}_8 \cong \mathbb{Z}_8^\times \cong \mathbb{Z}_2 \times \mathbb{Z}_2
$, just notice that every one of the four elements of $\displaystyle \mathbb{Z}_8^\times$ has order 2. There is only one group of order 4 in which every element has order 2; it is the Klein 4-group.
Notice that an homomorphism $\displaystyle f:G \rightarrow H$ when $\displaystyle G$ is cyclic is completely determined once you know the image of a generator. Knowing this $\displaystyle g: \mathbb{Z}_p \rightarrow \mathbb{Z}_p$ is determined by $\displaystyle g(1)$, but if $\displaystyle g$ is to be an automorphism then $\displaystyle g(1)=a$ must be a generator ie. a nonzero element, but we have $\displaystyle p-1$ such elements. Now take $\displaystyle g,h : \mathbb{Z}_p \rightarrow \mathbb{Z}_p$ automorphisms then if $\displaystyle a=h(1), b=g(1)$ then $\displaystyle gh(1)=ba$ so we can identify an automorphism $\displaystyle g$ with it's value at $\displaystyle 1$ and this defines an isomorphism between $\displaystyle Aut( \mathbb{Z}_p ) \cong \mathbb{Z}_p^{\times }$
Yes, that is how one goes about showing that $\displaystyle
\mbox{Aut } \mathbb{Z}_p \cong \mathbb{Z}_p^\times
$. However, it is showing that $\displaystyle \mathbb{Z}_p^\times \cong \mathbb{Z}_{p-1}$ which is difficult (and which I doubt has to be proven by mms - it is probably to be assumed).
Jose explained it, but I can explain some more.
For every $\displaystyle a \in \mathbb{Z}_p^\times$, let $\displaystyle \sigma_a : \mathbb{Z}_p \rightarrow \mathbb{Z}_p$ be the automorphism $\displaystyle x \mapsto ax$ (prove that it is an automorphism). Then the map $\displaystyle \pi : \mathbb{Z}_p^\times \rightarrow \mbox{Aut }\mathbb{Z}_p$ defined by $\displaystyle a \mapsto \sigma_a$ is the isomorphism you are looking for. (Prove it!)