Powers of Group Elements/Sum of Indices

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Theorem

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

Let $g \in G$.


Then:

$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$


Additive Notation

This can also be written in additive notation as:

$\forall m, n \in \Z: m g + n g = \paren {m + n} g$


Proof

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

$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$

$\blacksquare$


Also see


Sources