Size of conjugacy class divides order of inner automorphism group

Statement

Suppose $G$ is a group. Then, for any Conjugacy class (?) $c$ of $G$:

1. The size of $c$ is not greater than the order of the Inner automorphism group (?) (or equivalently, the index of the 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.

Related facts

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.