After proving that an element and its inverse have the same order, we now study exactly which powers of an element are equal to the identity. If $o(a)=n$, then $a^n=e$, but many other powers may also be equal to $e$. The precise rule is divisibility: $a^p=e$ exactly when $n$ divides $p$. This theorem is one of the most important tools for working with orders because it turns a group equation into an arithmetic divisibility statement. In this lesson, students will prove this criterion and use it to decide when powers equal the identity.

Let $(G,\circ)$ be a group with identity element $e$ and let $a\in G$. If $o(a)=n$, then for every positive integer $p$, $$\begin{equation*} a^p=e \iff n\mid p. \end{equation*}$$