Conjugacy Classes — Mathematics Knowledge System

Overview

Two elements a and b of a group G are conjugate if b = gag⁻¹ for some g ∈ G, and this relation partitions G into disjoint conjugacy classes — sets of elements that are 'the same up to a change of viewpoint.' Conjugacy classes are central to understanding a group's structure: a subgroup is normal exactly when it is a union of conjugacy classes, the class equation constrains possible group orders (it's the key tool in proving nontrivial centers for p-groups), and in the symmetric group Sₙ, conjugacy classes correspond exactly to cycle types, making them fully computable by counting cycle-shape partitions of n.

Intuition

Conjugation gag⁻¹ can be read as 'do g's perspective, then a, then undo g's perspective' — it's how element a looks when viewed through the lens of g. Two elements in the same conjugacy class behave identically as far as the group's abstract structure is concerned; only labels differ. In the symmetric group this becomes vivid: conjugating a permutation just relabels which objects get cycled, so two permutations are conjugate in Sₙ precisely when they have the same cycle type (same shape of cycles, ignoring which specific numbers appear in them) — turning an abstract equivalence relation into pure combinatorics of partitions of n.

Formal Definition

Definition

For a group G, define conjugation of a by g as gag⁻¹. Conjugacy is an equivalence relation, partitioning G into conjugacy classes:

a \sim b \iff \exists\, g \in G:\ b = gag^{-1}

Conjugacy relation

\text{Cl}(a) = \{ gag^{-1} : g \in G \}

Conjugacy class of a

|G| = |Z(G)| + \sum_i [G : C_G(x_i)]

Class equation (sum over non-central class representatives x_i)

|\text{Cl}(a)| = [G : C_G(a)], \quad C_G(a) = \{g \in G : ga = ag\}

Orbit–stabilizer: class size equals index of the centralizer

Theorems

Theorem 1: Conjugacy classes in Sₙ ↔ cycle types

\text{Two permutations in } S_n \text{ are conjugate if and only if they have the same cycle type (same partition of } n\text{).}

Theorem 2: Normal subgroups are unions of conjugacy classes

\text{A subgroup } N \trianglelefteq G \text{ if and only if } N \text{ is a union of complete conjugacy classes of } G.

Worked Examples

The identity is always alone in its own class, since geg⁻¹ = e for all g.
$\text{Cl}(e) = \{e\}$
Conjugating a transposition just relabels which two points it swaps; all three transpositions have the same cycle type (2,1), so they form one class.
$\text{Cl}((12)) = \{(12), (13), (23)\}$
The two 3-cycles share cycle type (3); conjugating (123) by any element yields either (123) or (132), giving the third class.
$\text{Cl}((123)) = \{(123), (132)\}$

Answer: S₃ has 3 conjugacy classes: {e} (size 1), {(12),(13),(23)} (size 3), {(123),(132)} (size 2) — sizes summing to 6 = |S₃|.

Practice Problems

Difficulty 5/10

List the conjugacy classes of S₄ by cycle type, and give the size of each class. (|S₄| = 24.)

Difficulty 5/10

In D₄ = ⟨r,s | r⁴,s²,srsr⟩ (order 8), compute the conjugacy class of r (a 90° rotation), given that s r s⁻¹ = r⁻¹ = r³.

Difficulty 6/10

Explain why, for any group G, the conjugacy class of the identity element e is always {e} alone — no matter what G is.

Common Mistakes

Common Mistake

Assuming conjugate elements are always equal or that conjugacy classes must be subgroups.

Conjugacy classes are generally NOT subgroups (they don't contain the identity unless the class IS {e}, and aren't closed under the group operation); they are equivalence classes under the conjugation relation, and only unions of them can form normal subgroups.

Quiz

Summary

Conjugacy classes partition G via a ~ b iff b = gag⁻¹ for some g; class size equals [G : C_G(a)] by orbit–stabilizer.
In Sₙ, conjugacy classes correspond exactly to cycle types — e.g. S₃ has classes {e}, {transpositions} (size 3), {3-cycles} (size 2).
The class equation |G| = |Z(G)| + Σ[G:C_G(xᵢ)] is the key tool for structural results like nontrivial centers of p-groups, and normal subgroups are precisely unions of conjugacy classes.

References

WebsiteWikipedia — Conjugacy class