Powers of Group Elements/Product of Indices

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Theorem

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

Let $g \in G$.


Then:

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


Additive Notation

This can also be written in additive notation as:

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


Proof

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

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

$\blacksquare$


Also see


Sources