Automorphism Maps Generator to Generator

Theorem
Let $G$ be a cyclic group.

Let $g$ be a generator of $G$.

Let $\phi$ be an automorphism on $G$.

Then $\map \phi g$ is also a generator of $G$.

Proof
By definition of automorphism, $\phi$ is a homomorphism

It follows that this result is a specific instance of Homomorphic Image of Cyclic Group is Cyclic Group.