The order of an element divides the order of a finite group. This divisibility has an important power consequence. If $a$ is an element of a finite group $G$, then raising $a$ to the power $|G|$ gives the identity element. This result comes directly from the cyclic subgroup generated by $a$. In this lecture, we prove the formula, compute examples, and clarify how the result depends on the finite nature of the group.

Let $(G,\circ)$ be a finite group with identity element $e$. If $a\in G$, then $$\begin{equation*} a^{|G|}=e. \end{equation*}$$