# Powers of Group Elements/Sum of Indices

## Theorem

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

Let $g \in G$.

Then:

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

This can also be written in additive notation as:

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

## Proof

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

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

$\blacksquare$