Inverse of Generator of Cyclic Group is Generator/Proof 1
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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 $\gen g = G$.
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.
$\blacksquare$