Conjugacy class

Conjugation partitions a group into equivalence classes. The sizes of these classes are a coarse structural fingerprint of the group and are the starting point for character theory.

Definition

Elements a, b ∈ G are conjugate, a ~ b, if there exists g ∈ G with b = gag⁻¹. This is an equivalence relation. The equivalence classes are the conjugacy classes of G; write aG for the class of a.

Two closely related subgroups:

• Centralizer of a ∈ G: CG(a) = {g ∈ G : ga = ag}.
• Normalizer of H ≤ G: NG(H) = {g ∈ G : gHg⁻¹ = H}.

CG(a) is the stabilizer of a under conjugation; NG(H) is the stabilizer of H under conjugation on subgroups.

Size of a class

By the orbit-stabilizer theorem applied to the conjugation action:

|aG|  =  [G : CG(a)].

In particular, |aG| divides |G|, and aG = {a} iff a ∈ Z(G) (see center).

Class equation

Let Z(G) be the center, and pick representatives a₁, …, aₖ of the non-central conjugacy classes. Then

|G|  =  |Z(G)|  +  Σᵢ [G : CG(aᵢ)].

Every summand on the right is > 1 and divides |G|. This identity is the engine behind many structural results — for instance, every p-group has a non-trivial center (see p-group).

Examples

• Abelian groups. Every class is a singleton: |aG| = 1 for all a.
• Symmetric group. Conjugacy classes of Sₙ are indexed by partitions of n (i.e. cycle types). S₄ has 5 classes, sizes 1, 3, 6, 6, 8, summing to 24.
• Dihedral group D₄. Classes: {e}, {r²} (central), {r, r³}, {s, r²s}, {rs, r³s}. Sizes 1, 1, 2, 2, 2.
• Quaternion group Q₈. Classes: {1}, {−1}, {±i}, {±j}, {±k}. Sizes 1, 1, 2, 2, 2.

In both D₄ and Q₈ the class sizes coincide, yet the groups are not isomorphic — class sizes alone don't identify a group.

Properties

1. Normal subgroups are unions of classes. This is equivalent to the definition of normal.
2. |aG| divides |G|. (Orbit-stabilizer.)
3. Number of classes equals number of complex irreducible representations. The foundational statement of character theory.
4. aG = {a} iff a ∈ Z(G).

Computation

• Class enumeration in a polycyclic group. Enumerate by walking the pc series and tracking conjugate-closures; scales with the number of classes times the number of generators.
• Class enumeration in a permutation group. Use random conjugations with product replacement to sample, combined with orbit-stabilizer to compute class sizes; for small groups, direct enumeration by BFS on conjugates works.
• Centralizer CG(a). Compute as the kernel of the map g ↦ [g, a]; polynomial in the number of generators.

See also

• Group action — orbit-stabilizer drives the class equation.
• Center — the union of singleton classes.
• Normal subgroup — unions of classes.
• Character table — one row per class, one column per irreducible representation.
• Wikipedia: Conjugacy class