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