Inverse of Generator of Cyclic Group is Generator/Proof 2

Theorem
Let $\left \langle {g} \right \rangle = G$ be a cyclic group.

Then $G = \left \langle {g^{-1}} \right \rangle$.

Thus, in general, the generator of a cyclic group is not unique.

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:
 * $g^{n-1} = g^{-1}$