Inverse of Generator of Cyclic Group is Generator/Proof 2

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 $C_n = \gen g$ 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 Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order that:

$C_n = \gen {g^{n - 1} }$

But from Inverse Element is Power of Order Less 1:

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

$\blacksquare$

Also see

• Inverse of Generator of Cyclic Group is Generator/Proof 1: note that from Inverse Element is Power of Order Less 1:

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