After proving that the identity element of a group is unique, we next prove that the inverse of each element is also unique. This is one of the most frequently used facts in group theory. The group definition says that for each $a\in G$, there exists an inverse element $a^{-1}\in G$, but the definition does not initially say that the inverse is the only one. The uniqueness theorem allows us to write $a^{-1}$ without ambiguity. In this lesson, students will learn the formal proof of uniqueness of inverses, how inverse notation changes in additive and multiplicative groups, and why both left and right inverse equations are essential.

Let $(G,\circ)$ be a group with identity element $e$, and let $a\in G$. An element $b\in G$ is called an $\textbf{inverse element}$ of $a$ if $$\begin{equation*} a\circ b=b\circ a=e. \end{equation*}$$ When the inverse element is unique, it is denoted by $a^{-1}$.