
In
mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, especially
group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, two elements
and
of a
group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
are conjugate if there is an element
in the group such that
This is an
equivalence relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
whose
equivalence class
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
es are called conjugacy classes.
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 group
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (mathematics), group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that '' ...
s is fundamental for the study of their structure.
For an
abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, each conjugacy class is a
set containing one element (
singleton set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
).
Function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 group representation, representation theory of groups, a class function is a function (mathematics), function on a group (mathematics), group ''G'' that is constant on the conjugacy class ...
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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
of
invertible matrices
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...
, 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
In mathematics
Mathematics (from Ancient ...
.
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
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
and
are conjugate, and
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:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
.
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
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
they can be described by cycle structure.
Examples
The symmetric group
consisting of the 6
permutation
In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

s of three elements, has three conjugacy classes:
# no change
#
transposing two
# a
cyclic permutation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of all three
These three classes also correspond to the classification of the
isometries
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of an
equilateral triangle
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

.

The symmetric group
consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their cycle structures and orders:
:(1)
4 no change (1 element: ). The single row containing this conjugacy class is shown as a row of black circles in the adjacent table.
:(2) interchanging two (6 elements: ). The 6 rows containing this conjugacy class are highlighted in green in the adjacent table.
:(3) a cyclic permutation of three (8 elements: ). The 8 rows containing this conjugacy class are shown with normal print (no boldface or color highlighting) in the adjacent table.
:(4) a cyclic permutation of all four (6 elements: ). The 6 rows containing this conjugacy class are highlighted in orange in the adjacent table.
:(2)(2) interchanging two, and also the other two (3 elements: ). 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 partition
300px, Partitions of ''n'' with biggest addend ''k''
In number theory and combinatorics
Combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mat ...
s 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
can be studied by
conjugation of isometries in Euclidean space.
Properties
* The identity element is always the only element in its class, that is
* If
is
abelian 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:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
. More generally, every statement about
can be translated into a statement about
because the map
is an
automorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of
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)
In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
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 (mathematics), 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 ...
consisting of all elements
such that
then the
index
Index 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 megastructure in the ''Halo'' series ...