Powers of Group Elements
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Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $g \in G$.
Then the following results hold:
Negative Index
- $\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$
Sum of Indices
- $\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$
Product of Indices
- $\forall m, n \in \Z: \paren {g^m}^n = g^{m n} = \paren {g^n}^m$