Conjugacy Class - Definition

Definition

Suppose G is a group. Two elements a and b of G are called conjugate if there exists an element g in G with

gag−1 = b.

(In linear algebra, this is referred to as matrix similarity.)

It can be readily shown that conjugacy is an equivalence relation and therefore partitions G into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes Cl(a) and Cl(b) are equal if and only if a and b are conjugate, and disjoint otherwise.) The equivalence class that contains the element a in G is

Cl(a) = { gag−1: g ∈ G }

and is called the conjugacy class of a. The class number of G is the number of distinct (nonequivalent) conjugacy classes.

Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class of order 6 elements", and "6B" would be a different conjugacy class of order 6 elements; the conjugacy class 1A is the conjugacy class of the identity. In some cases, conjugacy classes can be described in a uniform way – for example, in the symmetric group they can be described by cycle structure.