Inverse of Generator of Cyclic Group is Generator

From ProofWiki
Jump to: navigation, search

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 1

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.

$\blacksquare$


Proof 2

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:

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

$\blacksquare$


Sources