text: 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
  
  $g^{n - 1} = g^{-1}$
explanation: The provided text is a mathematical theorem and proof related to cyclic groups. It discusses the property that the inverse of a generator of a cyclic group is also a generator. The text does not contain a question or an answer to a question.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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
  
  $g^{n - 1} = g^{-1}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a mathematical theorem and proof related to cyclic groups. It discusses the property that the inverse of a generator of a cyclic group is also a generator. The text does not contain a question or an answer to a question.
  
  Label: no-question