Number of conjugacy classes: Difference between revisions

Revision as of 22:28, 6 April 2010

This article defines an arithmetic function on groups
View other such arithmetic functions

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

@@ Line 9: / Line 9: @@
 * 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 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 representation]]s.
+* The number of conjugacy classes in a [[finite group]] equals the number of equivalence classes of [[irreducible representation]]s. {{further|[[number of irreducible representations equals number of conjugacy classes]]}}