Powers of Group Elements/Negative Index/Additive Notation

Theorem
Let $\struct {G, +}$ be a group whose identity is $e$.

Let $g \in G$.

Then:
 * $\forall n \in \Z: -\paren {n g} = \paren {-n} g = n \paren {-g}$

Proof
All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Sum of Indices:


 * $\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$

where in this context:
 * the group operation is $+$
 * the $n$th power of $g$ is denoted $n g$
 * the inverse of $g$ is $-g$.