Powers of Group Elements/Negative Index/Additive Notation
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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 $-g$.
$\blacksquare$
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.1$: Subrings: Notation $2$