Size of conjugacy class divides order of inner automorphism group

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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

Analogous facts about degrees of irreducible representations

Applications

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

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