Proof that the order of an element divides the order of the group

Proof

1. Let $G$ be a finite group and let $g \in G$.
2. The cyclic subgroup $\langle g \rangle$ is a subgroup of $G$.
3. The order of $\langle g \rangle$ is the order of $g$.
4. By Lagrange's theorem, $|\langle g \rangle|$ divides $|G|$.
5. Thus the order of $g$ divides $|G|$.

Idea

The element generates a subgroup whose order is the order of the element. Lagrange's theorem then gives the result.