Order of Element Divides Order of Finite Group

Theorem
In a finite group, the order of a group element divides the order of its group:


 * $\forall x \in G: \left\vert{x}\right\vert \mathrel \backslash \left\vert{G}\right\vert$

Proof
Let $G$ be a group.

Let $x \in G$.

By definition, the order of $x$ is the order of the subgroup generated by $x$.

Therefore, by Lagrange's Theorem, $\left\vert{x}\right\vert$ is a divisor of $\left\vert{G}\right\vert$.

Also see

 * Element to Power of Group Order is Identity