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
This follows directly from Set of Words Generates Group. If $$\left \langle {g} \right \rangle = G$$ then therefore $$W \left({\left\{{g, g^{-1}}\right\}}\right) = G$$.

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