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:


 * $\forall x \in G: \left|{x}\right| \mathop \backslash \left|{G}\right|$

Proof
Let $G$ be a group.

Let $x \in G$.

By Order of Subgroup Generated by Single Element, the order of the subgroup generated by $x$ equals the order of $x$.

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

Also see

 * Element to the Power of Group Order in which it is shown that $x^{\left|{G}\right|} = e$.