Size of conjugacy class divides order of inner automorphism group

Statement
Suppose $$G$$ is a group. Then, for any fact about::conjugacy class $$c$$ of $$G$$:


 * 1) The size of $$c$$ is not greater than the order of the fact about::inner automorphism group (or equivalently, the index of the fact about::center). This statement is true both when the latter is finite and when both are infinite, where it is interpreted in terms of infinite cardinals.
 * 2) In the case that the order of the inner automorphism group is finite, the size of $$c$$ divides this order.
 * 3) When the order of the inner automorphism group is finite and greater than $$1$$, the size of $$c$$ is strictly smaller than the order of the inner automorphism group.

Similar facts

 * Size of conjugacy class equals index of centralizer
 * Size of conjugacy class divides order of group

Analogous facts about degrees of irreducible representations

 * Degree of irreducible representation divides order of inner automorphism group
 * Order of inner automorphism group bounds square of degree of irreducible representation

Applications

 * FZ implies FC: This states that if the center has finite index, then every conjugacy class has finite size.

Other related facts

 * Cyclic over central implies abelian: This is a very similar fact, which justifies the fact that the center cannot be a maximal subgroup.