Size of conjugacy class divides order of group

Statement
Let $$c$$ be a fact about::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 $$G$$.

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) uses::Size of conjugacy class equals index of centralizer