Number of conjugacy classes: Difference between revisions
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==Definition== | ==Definition== | ||
The '''number of conjugacy classes''' in a group is the number of [[conjugacy | The '''number of conjugacy classes''' in a group is the number of [[defining ingredient::conjugacy class]]es, viz the number of equivalence classes under the equivalence relation of being [[defining ingredient::conjugate elements|conjugate]]. | ||
==Facts== | ==Facts== | ||
Revision as of 00:28, 28 September 2010
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