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:


 * $\forall x \in G: \order x \divides \order G$

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, $\order x$ is a divisor of $\order G$.

Also see

 * Element to Power of Group Order is Identity