Size of conjugacy class divides order of group
From Groupprops
Contents
Statement
Let be a Conjugacy class (?) in a group
. The following are true:
- If
is a finite group, the size of
divides the order of
.
- In general, the size of
is not greater (as a cardinal) than the size of
.
Related facts
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 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.