Show that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+) $ holds

Question

I want to show, that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+)$ holds with $p$ prime.
($\mathbb{F}_p^+$ is the additive group, $\mathbb{F}_p^\times$ multiplicative group)

As hint we got the following homomorphism $$ \Phi :\mathbb{F}_p^+ \rightarrow \text{Aut}(\mathbb{F}_p^+), \ g \mapsto (x \mapsto g+x+(-g)) $$ $\text{kern}(\Phi)=\mathbb{F}_p^+$ because $\mathbb{F}_p^+$ is abelian and $\text{im}(\Phi)=\{\text{id}\}$.

I thought to create another homomorphism $$ \Phi' : \mathbb{F}_p^\times \rightarrow \text{Aut}(\mathbb{F}_p^+), \ g \mapsto (x \mapsto g \cdot x) $$ It is clear that $\text{kern}(\Phi')=\{\overline{1}\}$ and it follows $\mathbb{F}_p^ \times \simeq \text{im}(\Phi') \subseteq \text{Aut}(\mathbb{F}_p^+)$.

How do I know, that there are not more automorphisms?

Answer 1

HINT: The additive group $\Bbb F_p$ is cyclic, generated by any non-zero class. Now, an automorphism of $\Bbb F_p$ is a homomorphism $$ f:\Bbb F_p\longrightarrow\Bbb F_p $$ and a homomorphism between cyclic groups is completely determined by the image of a single chosen generator.

This should convince you that the map $\Phi^\prime$ you defined is surjective.

Answer 2

$\mathbf{F}_p$ is a cyclic group. You know a lot about group homomorphisms whose domain is a cyclic group.