Inverse of Generator of Cyclic Group is Generator/Proof 1

Proof
Let $\left\langle{g}\right\rangle = G$.

Then from Set of Words Generates Group:
 * $W \left({\left\{{g, g^{-1}}\right\}}\right) = G$

But:
 * $\left\langle {g^{-1}}\right\rangle = W \left({\left\{{g, g^{-1} }\right\}}\right)$

and the result follows.