Size of conjugacy class divides order of group

Statement

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

1. If $G$ is a finite group, the size of $c$ divides the order of $G$.
2. In general, the size of $c$ is not greater (as a cardinal) than the size of $G$.

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 $p$ 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.

Facts used

1. Size of conjugacy class equals index of centralizer