Inverse of Generator of Cyclic Group is Generator/Proof 2

Proof
Let $C_n = \gen g$ be the cyclic group of order $n$.

By definition, $g^n = e$.

We have that $n - 1$ is coprime to $n$.

So it follows from that Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order that:
 * $C_n = \gen {g^{n - 1} }$

But from Inverse Element is Power of Order Less 1:
 * $g^{n - 1} = g^{-1}$

Also see

 * Inverse of Generator of Cyclic Group is Generator/Proof 1: note that from Inverse Element is Power of Order Less 1:
 * $g^{n - 1} = g^{-1}$