Conjugacy class

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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.[1][2] In all abelian groups every 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.

Contents

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

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

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

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.

Examples

The symmetric group S3, consisting of all 6 permutations of three elements, has three conjugacy classes:

  • no change (abc → abc)
  • interchanging two (abc → acb, abc → bac, abc → cba)
  • a cyclic permutation of all three (abc → bca, abc → cab)

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