Automorphism Maps Generator to Generator
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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.
$\blacksquare$