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:

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

$\blacksquare$

Also see

$g^{n - 1} = g^{-1}$
The text passage is a theorem and proof about the properties of cyclic groups, specifically that the inverse of a generator of a cyclic group is also a generator. It does not contain any mathematical questions or answers to such questions.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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:

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

$\blacksquare$

Also see

$g^{n - 1} = g^{-1}$
user
The text passage is a theorem and proof about the properties of cyclic groups, specifically that the inverse of a generator of a cyclic group is also a generator. It does not contain any mathematical questions or answers to such questions.

TYPE: NO_QUESTION