Count symmetry inside the group itself

Try

Relabel the transposition (1 2) using a permutation that sends labels one and two to two and three. Conjugation produces (2 3). The labels change, while the cycle structure remains the same. Conjugation describes an element as seen after changing the surrounding coordinates.

Define g·x=gxg⁻¹. Check that identity acts trivially and that acting by h then g equals acting by gh. This verifies an actual group action rather than relying on a geometric analogy.

Predict

Which elements x have singleton conjugacy classes? The condition gxg⁻¹=x for every g is equivalent to gx=xg for every g. Predict the connection to the center Z(G), the subgroup of elements commuting with the entire group.

Experiment

List the conjugacy classes in S₃. Conjugation relabels cycles, so all transpositions form one class and both 3-cycles form another.

Representative	Class size	Centralizer size
e	1	6
(1 2)	3	2
(1 2 3)	2	3

Each row satisfies class size times centralizer size equals six. The class sizes sum to six, and only the identity has a singleton class, so the center is trivial. Verify one nontrivial conjugate directly to connect the relabeling rule with the product formula.

Understand

The centralizer C_G(x) is the set of g commuting with x. It is exactly the stabilizer of x under conjugation, since gxg⁻¹=x if and only if gx=xg. Orbit-stabilizer therefore gives the conjugacy-class size [G:C_G(x)]. Centralizers concern one chosen element; the center is the intersection of all centralizers.

Conjugacy classes partition G. For finite G, separate the singleton classes from the others to obtain |G|=|Z(G)|+Σ[G:C_G(xᵢ)], where one xᵢ represents each noncentral class. Taking one representative per class is essential; summing over every noncentral element would count each class repeatedly.

A subgroup N is normal precisely when it is a union of whole conjugacy classes. Normality means every conjugate of each member remains in N. But a union of classes need not be a subgroup: in S₃ the identity together with all three transpositions is conjugation-invariant, yet the product of two distinct transpositions is a missing 3-cycle. Closure remains a separate requirement.

This distinction can rule out subgroups. In A₄ the conjugacy-class sizes are 1,3,4,4: identity, the three double transpositions, and two classes of 3-cycles. For a 3-cycle its centralizer in A₄ has order three, giving class size four; for a double transposition its centralizer has order four, giving class size three. A subgroup of order six would have index two and be normal. But no sum of these class sizes including the identity equals six. Thus A₄ has no subgroup of order six, completing a counterexample to the converse of Lagrange.

Apply

Explain why equal class sizes do not mean equal classes. Then compute one centralizer directly and compare its size with the orbit formula. This provides two independent routes to the same count.

When using a class-sum obstruction, state both the subgroup’s required normality and the complete class list. A partial list or an arbitrary subgroup without normality does not justify summing whole classes. The proof’s force comes from combining those hypotheses.

Continue

For a group whose order is a prime power, every noncentral class size is divisible by that prime. The class equation then forces a nontrivial center and begins a structural classification.