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 $-g$.

$\blacksquare$