Inverse of Generator of Cyclic Group is Generator

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 1
This follows directly from Set of Words Generates Group.

If $\left \langle {g} \right \rangle = G$ then it follows that $W \left({\left\{{g, g^{-1}}\right\}}\right) = G$.

But of course $\left \langle {g^{-1}} \right \rangle = W \left({\left\{{g, g^{-1}}\right\}}\right)$ and the result follows.

Proof 2
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 Cyclic Group Element Coprime with Order is Generator‎ that $C_n = \left \langle {g^{n-1}} \right \rangle$.

This can be seen to be consistent with Proof 1, by noting that from Inverse Element is Power of Order Less 1, we have that $g^{n-1} = g^{-1}$.