Conjugacy class

From Online Dictionary of Crystallography

Classe de conjugaison (Fr). Konjugiertenklasse (Ge). Classe coniugata (It). 共役類 (Ja). Класс сопряжённости (Ru). Clase de conjugación (Sp).

If g1 and g2 are two elements of a group G, they are called conjugate if there exists an element g3 in G such that:

g3g1g3−1 = g2.

Conjugacy is an equivalence relation and therefore partitions G into equivalence classes: every element of the group belongs to precisely one conjugacy class.

The equivalence class that contains the element g1 in G is

Cl(g1) = { g3g1g3−1| g3G }

and is called the conjugacy class of g1. The class number of G is the number of conjugacy classes.

The classes Cl(g1) and Cl(g2) are equal if and only if g1 and g2 are conjugate, and disjoint otherwise.

For Abelian groups the concept is trivial, since each element forms a class on its own.