After proving the cancellation laws, we can study a useful consequence: the behaviour of idempotent elements in a group. An idempotent element is an element that remains unchanged when it is combined with itself. In many algebraic systems, there may be several idempotent elements. In a group, however, the cancellation laws force a very strong conclusion: the identity element is the only idempotent element. In this lesson, students will learn the definition of idempotent element, prove the group theorem, and compare the group case with familiar examples.

Let $(S,\circ)$ be a non-empty set with a binary operation $\circ$. An element $a\in S$ is called an $\textbf{idempotent element}$ if $$\begin{equation*} a\circ a=a. \end{equation*}$$ In multiplicative notation, this condition is written as $$\begin{equation*} a^2=a. \end{equation*}$$