In
mathematics, especially
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, two elements
and
of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
are conjugate if there is an element
in the group such that
This is an
equivalence relation whose
equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under
for all elements
in 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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
, each conjugacy class is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
containing one element (
singleton set).
Function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s that are constant for members of the same conjugacy class are called
class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjugat ...
s.
Definition
Let
be a group. Two elements
are conjugate if there exists an element
such that
in which case
is called of
and
is called a conjugate of
In the case of the
general linear group of
invertible matrices, the conjugacy relation is called
matrix similarity
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that
B = P^ A P .
Similar matrices represent the same linear map under two (possibly) different bases, with being ...
.
It can be easily shown that conjugacy is an equivalence relation and therefore partitions
into equivalence classes. (This means that 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 of
is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
.
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
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
.
Examples
The symmetric group
consisting of the 6
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
s of three elements, has three conjugacy classes:
# No change
. The single member has order 1.
#
Transposing two
. The 3 members all have order 2.
# A
cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ...
of all three
. The 2 members both have order 3.
These three classes also correspond to the classification of the
isometries of an
equilateral triangle.
The symmetric group
consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their description,
cycle type
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
, member order, and members:
# No change. Cycle type =
4">4 Order = 1. Members = . The single row containing this conjugacy class is shown as a row of black circles in the adjacent table.
# Interchanging two (other two remain unchanged). Cycle type =
221">221 Order = 2. Members = ). The 6 rows containing this conjugacy class are highlighted in green in the adjacent table.
# A cyclic permutation of three (other one remains unchanged). Cycle type =
131">131 Order = 3. Members = ). The 8 rows containing this conjugacy class are shown with normal print (no boldface or color highlighting) in the adjacent table.
# A cyclic permutation of all four. Cycle type =
1">1 Order = 4. Members = ). The 6 rows containing this conjugacy class are highlighted in orange in the adjacent table.
# Interchanging two, and also the other two. Cycle type =
2">2 Order = 2. Members = ). The 3 rows containing this conjugacy class are shown with boldface entries in the adjacent table.
The
proper rotations of the cube, which can be characterized by permutations of the body diagonals, are also described by conjugation in
In general, the number of conjugacy classes in the symmetric group
is equal to the number of
integer partitions of
This is because each conjugacy class corresponds to exactly one partition of
into
cycles, up to permutation of the elements of
In general, the
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations) ...
can be studied by
conjugation of isometries in Euclidean space
In a group, the conjugate by ''g'' of ''h'' is ''ghg''−1.
Translation
If ''h'' is a translation, then its conjugation by an isometry can be described as applying the isometry to the translation:
*the conjugation of a translation by a translation ...
.
Properties
* The identity element is always the only element in its class, that is
* If
is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
then
for all
, i.e.
for all
(and the converse is also true: if all conjugacy classes are singletons then
is abelian).
* If two elements
belong to the same conjugacy class (that is, if they are conjugate), then they have the same
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
. More generally, every statement about
can be translated into a statement about
because the map
is an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
of
called an
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
. See the next property for an example.
* If
and
are conjugate, then so are their powers
and
(Proof: if
then
) Thus taking
th powers gives a map on conjugacy classes, and one may consider which conjugacy classes are in its preimage. For example, in the symmetric group, the square of an element of type (3)(2) (a 3-cycle and a 2-cycle) is an element of type (3), therefore one of the power-up classes of (3) is the class (3)(2) (where
is a power-up class of
).
* An element
lies in the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of
if and only if its conjugacy class has only one element,
itself. More generally, if
denotes the of
i.e., the
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
consisting of all elements
such that
then the
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...