After defining a group, the next task is to understand the four group axioms separately. A group is not obtained by checking only one attractive property of an operation; all four conditions must work together. Closure keeps the result inside the set, associativity controls parentheses, the identity element leaves every element unchanged, and inverse elements bring each element back to the identity. These axioms appear simple, but many mistakes in abstract algebra begin from confusing one axiom with another. In this lesson, students will learn the exact role of each axiom and how to detect the failure of a group structure by locating the first failed axiom.

Let $G$ be a non-empty set and let $\circ$ be a rule on $G$. The set $G$ is said to be $\textbf{closed}$ under $\circ$ if $$\begin{equation*} a\circ b\in G \quad \forall\, a,b\in G. \end{equation*}$$ Closure means that the operation never takes two elements of $G$ outside $G$.