Conjugacy Classes in Point Groups

Introduction

The twelve operations of D₃h do not require twelve separate columns in a character table. Operations that are equivalent under a change of molecular orientation can be grouped into conjugacy classes. A class is defined algebraically, but it also has a geometric interpretation: different axes or planes can be related by a valid symmetry operation of the molecule. Understanding classes explains the compressed notation 2C₃ or 3σv and prepares the connection between class count and irreducible representations.

Core explanation

Two group operations A and B are conjugate if some operation X in the group satisfies B = XAX⁻¹. Read the right-hand product by applying X⁻¹ first, then A, then X. This sequence changes the reference orientation used to describe A. Conjugacy is an equivalence relation: every operation is conjugate to itself, the relation is symmetric, and it is transitive. The group therefore partitions into disjoint classes, each operation belonging to exactly one class.

In a molecular point group, symmetry-equivalent axes often produce conjugate rotations. For planar BF₃, a C₃ rotation can move one in-plane C₂′ axis into another. The 180° rotations about the three in-plane axes consequently lie in one class, written 3C₂′. Similarly the three vertical reflections lie in one 3σv class. A count preceding a symbol reports the number of operations in that class, not the power of the operation. D₃h has classes conventionally listed E, 2C₃, 3C₂′, σh, 2S₃ and 3σv.

Why do conjugate operations share a character? A representation maps each operation to a matrix. If B = XAX⁻¹, then D(B) = D(X)D(A)D(X)⁻¹. These matrices are similar, and similar matrices have the same trace. The trace is the character. Thus every operation within a conjugacy class has the same character in any representation, allowing one character-table column to stand for the whole class. The argument is linear algebra, not merely a naming convention.

The identity E forms a class by itself: XEX⁻¹ = E for every X. In an abelian group, every operation commutes with every other, so XAX⁻¹ = A; every class then contains one operation. Water's C₂v group is abelian and has four singleton classes for its four operations. Ammonia's C₃v group is non-abelian: C₃ and C₃² share a class, and its three mirror operations share another, giving three classes for six operations.

For a finite group, the number of irreducible representations equals the number of conjugacy classes. Hence C₂v has four irreducible species, C₃v has three and D₃h has six. This statement predicts the number of character-table rows, not the numerical value of each character. The dimensions dᵢ of irreducible representations satisfy the sum-of-squares relation Σdᵢ² = h, where h is group order. For C₃v, dimensions 1, 1 and 2 give 1² + 1² + 2² = 6. This is a useful consistency check.

An apparent similarity in geometry does not always imply conjugacy within the actual point group. Two rotations may have equal angle but act around axes that no group operation interchanges. Classes are determined by the group relation XAX⁻¹, not by outside visual judgement. Inversion, when present, often forms its own class because it commutes with all proper spatial operations, but the safest method remains the algebraic test.

Step-by-step reasoning

List the full group operations and fix a multiplication convention. For an operation A, calculate XAX⁻¹ for representative X values and collect the results. Repeat with any unassigned operation until all are partitioned. Count operations in each class and verify the class sizes sum to the group order. Only then compress operation headings into character-table columns.

Visual explanation

Draw an equilateral BF₃ triangle with three in-plane C₂′ axes through different B–F directions. A 120° C₃ turn carries the first axis into the second, and another turn carries it into the third. Circle the three axes as one class. Beside that, list C₂v's E, C₂ and two mirrors separately to show that visually similar mirrors need not merge in an abelian group.

Real-world analogy

Three doors in identical wings of a symmetric building are equivalent if an allowed building rotation takes one wing to another. A count of door types can then use one category with multiplicity three. Conjugacy formalises this equivalence for operations themselves, while a character table records one response per operation type.

Real-world example

When reducing a displacement representation for BF₃ vibrations, the same character is used for each of the three C₂′ operations, but its contribution to the reduction formula is multiplied by three. Forgetting the class size gives incorrect mode counts. Class notation makes a large group manageable while preserving the total number of operations in calculations.

Why?

Why does the number of classes equal the number of irreducible representations? Characters of irreducible representations form an orthogonal basis for functions that are constant on classes. The space of such class functions has one independent value per class, so a complete irreducible character table needs the same number of independent rows as class columns.

Common misconception

The expression 2C₃ does not mean a C₃ rotation performed twice; it means two distinct operations, C₃ and C₃², in one class. Also, equal characters in one particular representation do not prove two operations are conjugate. They must have the same characters in every representation, or satisfy the conjugation relation directly.

Worked example

For C₃v, list E, C₃, C₃² and three σv planes. Conjugation by a mirror reverses the sense of a C₃ turn, so C₃ and C₃² belong to one class. C₃ rotations move one vertical mirror into the next, so the three mirrors form a class. E remains alone. The class sizes 1, 2 and 3 sum to the group order 6; therefore its character table has three class columns and three irreducible rows.

Quick check

1. How many classes does an abelian group with four operations have? Answer: Four, because each operation forms a singleton class. 2. Why can one character describe all operations in a class? Answer: Their representation matrices are related by similarity transformations and therefore have equal traces.

Exam focus

Distinguish class size from rotation order and group order. Use the precise relation B = XAX⁻¹ when justifying conjugacy. Check that all class sizes sum to h, and include the multiplicity of each class in character reduction formulas.

Advanced insight

The sum-of-squares rule can rule out impossible character-table dimensions before any row is constructed. A six-operation group with three classes cannot have three one-dimensional irreducible representations because their squared dimensions would total only three, not six. The remaining dimension pattern 1, 1, 2 reflects the two-dimensional E species in C₃v.

Summary

Conjugacy groups operations into classes related by XAX⁻¹. All members of a class have the same character, so character tables use one column per class with its multiplicity noted. Abelian groups have singleton classes, while non-abelian groups can combine equivalent rotations or mirrors. Class counts and representation dimensions provide strong consistency checks.

Practice questions

1. A group has classes of sizes 1, 2 and 3. What is its order and how many irreducible representations does it have? Answer: Its order is 1 + 2 + 3 = 6, and it has three irreducible representations because it has three classes. 2. Why must E always be in its own conjugacy class? Answer: For every X, XEX⁻¹ = E, so conjugation never produces a different operation from identity. 3. In a character reduction sum, why multiply the character product for 3σv by three? Answer: The column represents three distinct operations, each contributing the same character product. Multiplicity preserves the full sum over every group operation.