Powers of Group Elements/Negative Index/Additive Notation

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Theorem

Let $\left({G, +}\right)$ be a group whose identity is $e$.

Let $g \in G$.


Then:

$\forall n \in \Z: - \left({n g}\right) = \left({-n}\right) g = n \left({-g}\right)$


Proof

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

$\forall n \in \Z: \left({g^n}\right)^{-1} = g^{-n} = \left({g^{-1} }\right)^n$

where in this context:

the group product operator is $+$
the $n$th power of $g$ is denoted $n g$
the inverse of $g$ is $-g$.

$\blacksquare$


Sources