Inverse of Generator of Cyclic Group is Generator/Proof 2

Proof
Let $C_n = \left\langle{g}\right\rangle$ 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 Element is Generator of Cyclic Group iff Coprime with Order‎ that:
 * $C_n = \left\langle{g^{n - 1} }\right \rangle$

Also see

 * Generator of Cyclic Group is not Unique/Proof 1: note that from Inverse Element is Power of Order Less 1:
 * $g^{n - 1} = g^{-1}$