# Size of conjugacy class divides order of group

From Groupprops

## Contents

## Statement

Let be a Conjugacy class (?) in a group . The following are true:

- If is a finite group, the size of divides the order of .
- In general, the size of is not greater (as a cardinal) than the size of .

## Related facts

### Stronger facts

- Size of conjugacy class divides order of inner automorphism group
- Size of conjugacy class equals index of centralizer

### Analogous facts about degrees of irreducible representations

- Degree of irreducible representation divides order of group
- Degree of irreducible representation divides order of inner automorphism group

## Related notions

The breadth of a finite p-group is defined as the logarithm to base of the smallest of the orders of centralizers of elements. The class-breadth conjecture is a conjectured inequality relating the breadth and the nilpotency class.