Injective homomorphism from $\mathbb Z/q \mathbb Z$ to $\text{Aut}(\mathbb Z/p \mathbb Z)$

Question

I am stuck in a complexer proof with the following "obvious" partial result:

Let $p$ and $q$ be distinct prime numbers with $q \mid p-1$. If $\text{Aut}(\mathbb Z/p \mathbb Z)$ is the automorphism group of $\mathbb Z/p \mathbb Z$, what concrete examples are there for injective group homomorphisms $\mathbb Z/q \mathbb Z \to \text{Aut}(\mathbb Z/p \mathbb Z)$ ?

I know that $|\text{Aut}(\mathbb Z/p \mathbb Z)| = p-1$ but have no idea how to construct the asserted injective group homomorphism.

This is not a homework exercise or something similiar.

Thank you for your thoughts!

Answer 1

One knows more about $Aut(\mathbb{Z}/p\mathbb{Z})$ than just the fact that its order is equal to $q$. One knows the group structure as well, namely, $Aut(\mathbb{Z}/p\mathbb{Z})$ is isomorphic to $\mathbb{Z}/p\mathbb{Z} - \{0\}$ under the operation of multiplication, which is isomorphic to the cyclic group of order $p-1$.

So, your question reduces to constructing all injective group homomorphisms $\mathbb{Z}/q\mathbb{Z} \to \mathbb{Z}/(p-1)\mathbb{Z}$, which in turn reduces to counting how many elements of $\mathbb{Z}/(p-1)\mathbb{Z}$ have order dividing $q$.