Conjugation (group theory)

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set).

Functions that are constant for members of the same conjugacy class are called class functions.

Let G be a group. Two elements a and b of G are conjugate, if there exists an element g in G such that gag⁻¹ = b. One says also that b is a conjugate of a and that a is a conjugate of b .

In the case of the group GL(n) of invertible matrices, the conjugacy relation is called matrix similarity.

It can be easily 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

and is called the conjugacy class of a. The class number of G is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same order.

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