## Conjugacy

Elements and of a group are called conjugate if there exists satisfying Conjugacy is an equivalence relation since

1. so 2. so 3. so  splits into equivalence classes Every element of the group belongs to precisely one conjugacy class, and the classes and are equal if and only if and are conjugate, and disjoint otherwise. The equivalence class that contains the element is and is called the conjugacy class of The class number of is the number of distinct (nonequivalent) conjugacy classes.

Elements of each conjugacy class have a similar structure. If the group elements act on a geometric object, elements of each conjugacy class have similar geometric effects. For example, all the rotations may form one class, all the reflections another, and the identity will be in a class by itself.

The symmetric group consisting of all 6 permutations of three labels, has three conjugacy classes:

• The identity (1)(2)(3).

• interchanging two two labels (12)(3), (13)(2),(1)(23).

• a cyclic permutation of all three labels (123), (132).

If is abelian, then for all so for all so conjugacy is not very useful in the abelian case. A subset of the group may be abelian so the conjugacy classes gives us an idea of the extent of non – abelianness.

If belong to the same conjugacy class - they are conjugate - then they have the same order and every statement about can be translated into a statement about because the map is an automorphism of An element lies in the center of if and only if its conjugacy class has only one element, a itself. More generally, if denotes the centralizer of i.e., the subgroup consisting of all elements such that then the index is equal to the number of elements in the conjugacy class of (by the orbit-stabilizer theorem).

If are conjugate, then so are powers of them, since  