After learning binary operation, we are ready to define the first central structure of abstract algebra. A group is a non-empty set together with a binary operation satisfying four structural conditions. These conditions are not arbitrary; they allow us to calculate, cancel, solve equations, discuss inverses, and compare algebraic systems that may look very different at first. Number systems under addition, nonzero number systems under multiplication, permutations under composition, and matrix groups all arise from this single definition. In this lesson, students will learn the formal definition of group, the roles of the identity element and inverse element, and the correct way to verify the definition in a basic example.

Let $G$ be a non-empty set and let $\circ$ be a binary operation on $G$. Then $(G,\circ)$ is called a $\textbf{group}$ if the following conditions hold: (i) $a\circ b\in G \quad \forall\, a,b\in G$. (ii) $(a\circ b)\circ c=a\circ(b\circ c) \quad \forall\, a,b,c\in G$. (iii) There exists $e\in G$ such that $$\begin{equation*} e\circ a=a\circ e=a \quad \forall\, a\in G. \end{equation*}$$ (iv) For each $a\in G$, there exists $a^{-1}\in G$ such that $$\begin{equation*} a\circ a^{-1}=a^{-1}\circ a=e. \end{equation*}$$ The element $e$ is called the $\textbf{identity element}$ of $G$, and $a^{-1}$ is called the $\textbf{inverse element}$ of $a$.