Since $\mathbb{Z}_{10}$ is a cyclic group (ex. generated by $\left<1\right>$), all automorphisms will be determined by finding $\phi(1)$. Since we want an isomorphism, we map $1$ to a generator, since then $\phi(1)$ will generate $\mathbb{Z}_{10}$. The generators of $\mathbb{Z}_{10}$ are numbers less than $10$, and co-prime with $10$. Thus $\phi(1) \in \{1,3,7,9\}$. Then the following will be automorphisms: $$\phi(x) = x \pmod{10}, \\ \phi_{3}(x) = 3x \pmod{10}, \\ \phi_{7}(x) = 7x \pmod{10},\\ \phi_{9}(x) = 9x \pmod{10}$$ Is this correct? Do I have to prove that each of these are isomorphisms(automorphisms)?
Update: I figured out that we can check if they are isomorphisms, since they are very simple mappings. So $\phi(x)$ is just the identity map, so that is clear. Now we can check $\phi_{3}(x)$. It is bijective. For the homomorphism:$\phi_{3}(x+y) = 3(x+y)= 3x + 3y = \phi_{3}(x) + \phi_{3}(y).$ The same goes for the others.