Inverse of Generator of Cyclic Group is Generator/Proof 1

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 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.