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:


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

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 $$\left|{G}\right|$$.