After studying powers of elements in a group, the next natural question is when those powers return to the identity element. This leads to the order of an element. The order of an element measures the first positive power that gives the identity, when such a power exists. This idea is central in abstract algebra because it connects powers, cyclic subgroups, roots of identity, and later results such as Lagrange’s theorem. In this lesson, students will learn finite order, infinite order, and how to compute the order of elements in familiar groups.

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 $\textbf{finite order}$. The least positive integer $n$ is called the $\textbf{order}$ of $a$ and is denoted by $$\begin{equation*} o(a)=n. \end{equation*}$$