# Element to Power of Group Order is Identity

## Theorem

Let $G$ be a group whose identity is $e$ and whose order is $n$.

Then:

$\forall g \in G: g^n = e$

## Proof

Let $G$ be a group such that $\order G = n$.

Let $g \in G$ and let $\order G = k$.

$k \divides n$

So:

$\exists m \in \Z_{>0}: k m = n$

Thus:

 $\ds g^n$ $=$ $\ds \paren {g^k}^m$ Powers of Group Elements: Product of Indices $\ds$ $=$ $\ds e^m$ Definition of Order of Group Element: $g^k = e$ $\ds$ $=$ $\ds e$ Power of Identity is Identity

$\blacksquare$