Size of conjugacy class divides order of group

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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

Analogous facts about degrees of irreducible representations

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