Lagrange's theorem restricts the orders of subgroups. Since every element $a$ of a group generates a cyclic subgroup $\langle a\rangle$, it also restricts the order of every element. This is one of the most useful consequences of Lagrange's theorem. Instead of studying $a$ alone, we study the subgroup generated by $a$. The size of this subgroup is exactly the order of the element, so it must divide the order of the whole group.

Let $(G,\circ)$ be a group with identity element $e$, and let $a\in G$. If there exists a least positive integer $n$ such that $$\begin{equation*} a^n=e, \end{equation*}$$ then $a$ is said to have $\textbf{finite order}$ and the order of $a$ is $$\begin{equation*} o(a)=n. \end{equation*}$$ If no such positive integer exists, then $a$ is said to have infinite order.