# Conjugacy class size statistics of a finite group

From Groupprops

## Contents

## Definition

Let be a finite group. The **conjugacy class size statistics** of is a function that outputs, for each , the number of conjugacy classes of of size . Note that since size of conjugacy class divides order of group, the function is nonzero only on (some) divisors of the order of .

The conjugacy class size statistics carry more information than the conjugacy class size set of a finite group, which is simply the set of sizes of the conjugacy classes in .

## Relation with other statistics

### Stronger statistics

- Conjugacy class root statistics of a finite group
- Conjugacy class-cum-order statistics of a finite group

## Facts

### Facts about conjugacy class sizes

Divisibility facts:

- Size of conjugacy class divides order of group
- Size of conjugacy class divides index of center
- Size of conjugacy class equals index of centralizer

Bounding facts:

Non-divisibility/non-bounding facts:

- Size of conjugacy class need not divide exponent
- Size of conjugacy class need not divide index of abelian normal subgroup
- Size of conjugacy class may be greater than index of abelian normal subgroup

### Relation with degrees of irreducible representations

The number of conjugacy classes is an important measure in relating conjugacy classes to irreducible representations and to the degrees of irreducible representations.