Conjugacy class - Wikipedia

Abstraction: Equivalence classes under group conjugation relation

Key points:

• Two elements a, b are conjugate if there exists g such that b = g·a·g⁻¹; this is an equivalence relation partitioning a group into conjugacy classes
• Members of the same conjugacy class share many properties and cannot be distinguished by group structure alone; identity element is always its own singleton class
• In abelian groups, every conjugacy class is a singleton; an element is in the center iff its class has size 1
• The class equation relates group order to center size and conjugacy class sizes; used to prove every finite p-group has nontrivial center
• In symmetric groups, conjugacy classes correspond exactly to cycle types (integer partitions of n); the number of classes equals the number of partitions
• Number of irreducible complex representations of any finite group equals its number of conjugacy classes (representation theory connection)

Connections: Wikipedia · Group Theory · Conjugacy Classes · Abstract Algebra

Source: http://en.wikipedia.org/wiki/Conjugacy_class