Combine invariants, actions and presentations

Try

Two groups can have the same size and radically different multiplication. A cyclic group of order six is abelian, while S₃ is not. To classify groups of order six, it is not enough to exhibit those two examples. We must prove every group of that order is isomorphic to one of them.

Begin with Sylow theory: the number of order-three subgroups divides two and is one modulo three, so exactly one exists. Call it ⟨a⟩.

Predict

Choose an element b of order two, whose existence follows from the Sylow theorem for p=2. Since ⟨a⟩ is normal, conjugation by b sends a to a generator of ⟨a⟩. Predict the only two possible images: a or a⁻¹. Each choice will determine a multiplication rule.

Experiment

The six forms aⁱbʲ, with i modulo three and j modulo two, are distinct and exhaust G. Distinctness follows because ⟨a⟩ and ⟨b⟩ have trivial intersection.

Conjugation rule	Product behavior	Resulting group
bab⁻¹=a	a and b commute	C₃×C₂≅C₆
bab⁻¹=a⁻¹	Moving b past a reverses exponent	D₃≅S₃

In the second case, map a to (1 2 3) and b to (1 2). These permutations satisfy the same relations and give six distinct forms. Matching the normal-form multiplication proves an isomorphism, rather than merely matching group orders.

Understand

A classification proof has three obligations: every object lies in a listed type, every listed type exists, and distinct listed types are not isomorphic. The preceding argument supplies completeness for order six. The two examples exist as familiar groups, and commutativity distinguishes them.

For order four, Lagrange permits nonidentity orders two or four. An element of order four gives C₄. Otherwise every nonidentity element is an involution. For distinct a,b, cancellation shows ab is neither e,a nor b, so it is the remaining nonidentity element. Since (ab)⁻¹=ab but also equals b⁻¹a⁻¹=ba, the group is abelian and is C₂×C₂. This derives commutativity rather than assuming it from the small order.

At order eight, several invariants are needed. The abelian types are C₈, C₄×C₂, and C₂³. Two nonabelian examples are D₄ and Q₈. In Q₈, the elements ±i,±j,±k all square to −1, and −1 is the only element of order two. In D₄ there are five involutions: four reflections and the half-turn. Hence these two groups are not isomorphic. This comparison distinguishes the examples; it is not by itself a proof that no further order-eight groups exist.

The quaternion relations i²=j²=k²=ijk=−1 determine the usual multiplication, with ij=k and ji=−k. A concrete realization by invertible complex matrices can verify associativity if constructing Q₈ from scratch. A collection of formal symbols and an asserted table requires such a consistency check.

Apply

Write the order-six proof as a decision tree with each branch justified by a theorem. Mark where existence is used, where normality is used, and where a map is constructed. Then identify an invariant separating the final branches.

For order twelve, compare C₁₂ with A₄. Both obey the same Lagrange divisibility condition, but A₄ is nonabelian and has no element of order twelve. Allowed divisors are constraints, not a multiplication law or a complete classification.

Continue

The final lesson combines these ideas in proof revision and unfamiliar cases. Keep a clear distinction between a theorem, a computed finite example, and a classification claim requiring an exhaustive argument.

The later complete order-eight classification supplies the exhaustion and consistency proofs missing from the comparison above. For a general distinct-prime result, use split extensions and groups of order pq.