In mathematics, especially group theory of a group are conjugate This is an equivalence relation whose equivalence classes are called conjugacy classes an element of the group.

Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set).

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

A normal subgroup is defined by the property that its conjugacy class contains a single member, namely itself. Normal subgroups play a key role in the study of quotient groups and group homomorphisms.

Motivation

The concept of conjugation may come from trying to formalize the idea that two group elements are considered the "same" after a relabeling of elements.

For example, consider the symmetric group on this new labeling.

have the same "shape". This includes their order, parity, degree, cycle type, etc.

is an automorphism to itself that preserves the group structure. Such an automorphism is called an inner automorphism.

Another way to illustrate the conjugacy action is by considering the general linear group of invertible matrices, which is the same condition as matrix similarity being the change-of-basis matrix.

complex matrix whose elements are the Cayley–Klein parameters, an operation that can also be expressed in terms of unit quaternions.

Conjugate subgroups come up in some important theorems of group theory. An example is one of the Sylow theorems are conjugates of each other.

Definition

are conjugate is called

In the case of the general linear group of invertible matrices, the conjugacy relation is called matrix similarity.

are equal if and only if are conjugate, and disjoint and is called the conjugacy class The 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 with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity which has order 1. In some cases, conjugacy classes can be described in a uniform way; for example, in the symmetric group they can be described by cycle type.

Properties

The identity element lies in the center is abelian is abelian).

belong to the same conjugacy class (that is, if they are conjugate), then they have the same order as noted previously, is an automorphism.

) Thus taking the ).

Examples

The symmetric group S_3, consisting of the 6 permutations of three elements, has three conjugacy classes:

1. Transposing

2. A cyclic permutation

These three classes also correspond to the classification of the isometries of an equilateral triangle.

For example, in the third class above (corresponding to rotations of the equilateral triangle by 120 and 240 degrees, respectively), a possible choice for g from the second class (corresponding to the flipping of the triangle around its axis going through a).

The symmetric group S_4, consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their members using cycle notation:

is equal to the number of integer partitions into cycles is

The dihedral group consisting of symmetries of a pentagon, has four conjugacy classes:

Conjugation as a group action, centralizers, and the class equation

This defines a group action The orbits denote the centralizer i.e., the subgroup Then the stabilizer

Conjugacy class equation

are in one-to-one correspondence with cosets This is a special case of the orbit-stabilizer theorem, keeping in mind that conjugacy classes are orbits and centralizers are stabilizers under the action of the group on itself through conjugation.

is a finite group is the index; hence the size of each conjugacy class divides the order of the group.

forms a conjugacy class containing just itself gives rise to the class equation: where the sum is over a representative element from each conjugacy class that is not in the center.

can often be used to gain information about the order of the center or of the conjugacy classes.

Example

Consider a finite p-group is a prime number-group has a center of size greater than 1.

is abelian and is, in fact, isomorphic either to a cyclic group

Average centralizer

By Burnside's lemma==Conjugacy of subgroups and general subsets==

More generally, given any subset

on the set of all subsets the normalizer are in the same coset equals the cardinality this formula specializes to the one given earlier for the number of elements in a conjugacy class since the normalizer of a singleton equals its centralizer.

The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate. Conjugate subgroups are isomorphic, but isomorphic subgroups need not be conjugate. For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate.

Geometric interpretation

Conjugacy classes in the fundamental group of a path-connected topological space can be thought of as equivalence classes of free loops under free homotopy.

Conjugacy class and irreducible representations in finite group

In any finite group, the number of nonisomorphic irreducible representations over the complex numbers is precisely the number of conjugacy classes.

See also

Notes

References

External links