Inverse of Generator of Cyclic Group is Generator/Proof 1

Theorem

Let $\gen g = G$ be a cyclic group.

Then:

$G = \gen {g^{-1} }$

where $g^{-1}$ denotes the inverse of $g$.

Thus, in general, a generator of a cyclic group is not unique.

Proof

Let $\gen g = G$.

Then from Set of Words Generates Group:

$\map W {\set {g, g^{-1} } } = G$

But:

$\gen {g^{-1} } = \map W {\set {g, g^{-1} } }$

and the result follows.

$\blacksquare$