After learning the definition and first examples of groups, we now begin the elementary properties that follow from the group axioms. These properties are not extra assumptions; they are consequences of closure, associativity, identity, and inverse. The first such property is the uniqueness of the identity element. This result is important because the group definition says that there exists an identity element, but it does not initially say that there is only one. In this lesson, students will learn why a group cannot have two different identity elements and how this uniqueness is used in later proofs involving inverses, cancellation, equations, and powers.

Let $(G,\circ)$ be a group. An element $e\in G$ is called the $\textbf{identity element}$ of $G$ if $$\begin{equation*} e\circ a=a\circ e=a \quad \forall\, a\in G. \end{equation*}$$ The identity element must leave every element of $G$ unchanged from both sides.