After studying power criteria for commutativity, we finish the abelian-groups section with elements of order two. Such elements are often called involutions. They are important because they are their own inverses and frequently control commutativity patterns in small groups. When two distinct elements of order two commute, their product also has order two. When an element of order two is unique in a group, it must commute with every element of the group. In this lesson, students will prove these order-two results carefully.

Let $(G,\circ)$ be a group with identity element $e$. An element $a\in G$ is called an $\textbf{element of order two}$ if $$\begin{equation*} o(a)=2. \end{equation*}$$ Equivalently, $$\begin{equation*} a\neq e \end{equation*}$$ and $$\begin{equation*} a^2=e. \end{equation*}$$