Given a generator for a cyclic subgroup of size d, is there a method to find any other of the $\varphi(d)-1$ generators for that subgroup?

Question

I'm given a generator for a subgroup, can I find any other generators?

Answer 1

If $G$ is cyclic and finite with generator $g$ then $g^a$ is a generator for all $a$ coprime to $\lvert G \rvert$,

because if $\gcd(a,\lvert G\rvert) = 1$ then there exist $m,n \in \mathbb{Z}$ with $ma + n\lvert G\rvert = 1$, so $$ (g^a)^m = g^{1-n\lvert G\rvert} = g.$$