Number of conjugacy classes

This article defines an arithmetic function on groups
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Definition

The number of conjugacy classes in a group is the number of conjugacy classes, viz the number of equivalence classes under the equivalence relation of being conjugate.

Facts

• A group with only finitely many conjugacy classes is termed an FC-group.
• The number of conjugacy classes in a group is at least as much as the number of elements in the center.
• The number of conjugacy classes is at most as much as the number of elements in the group. Equality (for FC-groups) holds if and only if the group is abelian.
• The number of conjugacy classes in a finite group equals the number of equivalence classes of irreducible representations. Further information: number of irreducible representations equals number of conjugacy classes