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 $$g^{n-1} = g^{-1}$$.