After studying the four group axioms separately, the next skill is verification. To prove that a given algebraic system is a group, we must check the four axioms in an organized order. This is not a matter of guessing from familiar examples. Each verification must mention the set, the operation, closure, associativity, identity, and inverses. This method is useful throughout abstract algebra because the same pattern appears later in subgroups, matrix groups, permutation groups, cyclic groups, and quotient groups. In this lesson, students will learn a reliable verification method for proving that a structure is a group.

The standard verification method has four steps. First, prove closure. Second, prove associativity. Third, identify the identity element and verify it from both sides. Fourth, for an arbitrary element of the set, find an inverse element that also belongs to the set. The word arbitrary is important: checking one or two elements is not enough unless the set is finite and all elements are checked. For infinite sets such as $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$, the proof must work for a general element.