Identify the operation before checking the axioms

Try

Take the residues zero through five and add modulo six. Every result stays in the set. Zero changes nothing, and each element has a partner whose sum is zero. Now use ordinary subtraction on the integers. Closure still holds, but compare (1−2)−3 with 1−(2−3). A familiar operation can fail the structure we want.

A group consists of a set together with an operation. Naming the set alone is incomplete: integers under addition form a group, while integers under multiplication do not. Record both pieces whenever you propose an example.

Predict

Do all groups require ab=ba? Predict whether composing two reversible rearrangements in opposite orders always has the same effect. Commutativity is an additional property, called abelian, rather than a group axiom. Associativity only changes parentheses while preserving the order of the factors.

Experiment

For addition modulo six, fill in the inverse pairs and check both orders of addition.

Element	Additive inverse	Sum modulo six
0	0	0
1	5	0
2	4	0
3	3	0

The remaining rows reverse the second and third pairs. Associativity comes from integer addition: reducing a sum modulo six at an intermediate step does not change its final residue. This is a reason valid for every triple, not merely a few matching calculations.

For subtraction, the two expressions in the opening are −4 and 2. One unequal triple disproves associativity. In contrast, a collection of equal triples would not prove it for infinitely many integers.

Understand

The group axioms require closure, associativity, a two-sided identity e, and a two-sided inverse for every element. Closure means the operation maps G×G into G. Associativity says (ab)c=a(bc). Identity means ea=ae=a. An inverse satisfies aa⁻¹=a⁻¹a=e. The notation is multiplicative even when the underlying operation is not numerical multiplication.

Identity is unique. If e and f are both identities, ef=f by the property of e, and ef=e by the property of f. Inverses are unique too: if b and c are inverses of a, then b=b(ac)=(ba)c=c. Notice the exact use of associativity; no factors were exchanged.

Cancellation follows by multiplying on the correct side. If ab=ac, multiply on the left by a⁻¹ and reassociate to obtain b=c. Right cancellation has its own analogous argument. The inverse of a product is (ab)⁻¹=b⁻¹a⁻¹, because ab b⁻¹a⁻¹=e and the reverse product also equals e. Reversing the order matters in nonabelian groups.

Nonzero real numbers under multiplication provide another group. All real numbers do not, because zero lacks a multiplicative inverse. Nonzero integers also fail: the inverse of two is not an integer. An inverse must belong to the specified set, not merely exist in a larger familiar number system.

Apply

Write an axiom audit for positive rational numbers under multiplication. Then change the set to positive integers and identify the precise failure. Avoid saying only that the second example “does not work.” Supply a member that has no allowed inverse.

In the practice, distinguish a proof for arbitrary elements from a counterexample. A proof must cover the whole definition; a counterexample needs only one decisive failure. For finite tables, associativity requires checking triples, not simply noticing a symmetric-looking arrangement.

Continue

Next identify smaller groups inside a known group. Inherited associativity simplifies the task, but a subset still needs identity, inverses and closure. The subgroup test combines those requirements into one efficient criterion.