Automorphism Maps Generator to Generator

Theorem
Let $G, H$ be cyclic groups.

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

Let $\varphi: G \to H$ be an isomorphism.

Then $\map \varphi g$ is a generator of $H$.

Proof
Therefore $H = \gen {\map \varphi g}$.

So $\map \varphi g$ is also a generator of $H$.