Conjugacy class
From Online Dictionary of Crystallography
Classe de conjugaison (Fr). Konjugationsklasse (Ge). Clase de conjugación (Sp). Classe coniugata (It). Класс сопряжённости (Ru). 共役類 (Ja).
If g_{1} and g_{2} are two elements of a group G, they are called conjugate if there exists an element g_{3} in G such that:
- g_{3}g_{1}g_{3}^{−1} = g_{2}.
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 g_{1} in G is
- Cl(g_{1}) = { g_{3}g_{1}g_{3}^{−1}| g_{3} ∈ G }
and is called the conjugacy class of g_{1}. The class number of G is the number of conjugacy classes.
The classes Cl(g_{1}) and Cl(g_{2}) are equal if and only if g_{1} and g_{2} are conjugate, and disjoint otherwise.
For Abelian groups the concept is trivial, since each element forms a class on its own.