TheInfoList

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 $a$ and $b$ 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 $g$ in the group such that $b = g^ag.$ 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 $G$ be a group. Two elements $a, b \in G$ are conjugate if there exists an element $g \in G$ such that $gag^ = b,$ in which case $b$ is called of $a$ and $a$ is called a conjugate of $b.$ 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 ...
$\operatorname\left(n\right)$ 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 $G$ into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes $\operatorname\left(a\right)$ and $\operatorname\left(b\right)$ 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 ...
$a$ and $b$ are conjugate, and otherwise.) The equivalence class that contains the element $a \in G$ is $\operatorname(a) = \left\$ and is called the conjugacy class of $a.$ The of $G$ 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 $S_3,$ 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 $\left(abc \to abc\right)$ # transposing two $\left(abc \to acb, abc \to bac, abc \to cba\right)$ # 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 $\left(abc \to bca, abc \to cab\right).$ 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 $S_4,$ 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 $S_4.$ In general, the number of conjugacy classes in the symmetric group $S_n$ 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 $n.$ 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 $\operatorname\left(e\right) = \.$ * If $G$ is abelian then $gag^ = a$ for all $a, g \in G$, i.e. $\operatorname\left(a\right) = \$ for all $a \in G$ (and the converse is also true: if all conjugacy classes are singletons then $G$ is abelian). * If two elements $a, b \in G$ 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 $a$ can be translated into a statement about $b = gag^,$ because the map $\varphi\left(x\right) = gxg^$ 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 $G.$ See the next property for an example. * If $a$ and $b$ are conjugate, then so are their powers $a^k$ and $b^k.$ (Proof: if $a = gbg^$ then $a^k = \left\left(gbg^\right\right)\left\left(gbg^\right\right) \cdots \left\left(gbg^\right\right) = gb^kg^.$) Thus taking $k$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 $a$ is a power-up class of $a^k$). * An element $a \in G$ 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 ...
$\operatorname\left(G\right)$ of $G$ if and only if its conjugacy class has only one element, $a$ itself. More generally, if $\operatorname_G\left(a\right)$ denotes the of $a \in G,$ 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 $g$ such that $ga = ag,$ 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 ...

# Conjugacy as group action

For any two elements $g, x \in G,$ let $g \cdot x := gxg^.$ This defines a
group action 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). ...
of $G$ on $G.$ The
orbits In celestial mechanics Celestial mechanics is the branch of astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical obj ...
of this action are the conjugacy classes, and the
stabilizer Stabilizer, stabiliser, stabilisation or stabilization may refer to: Chemistry and food processing * Stabilizer (chemistry), a substance added to prevent unwanted change in state of another substance ** Polymer stabilizers are stabilizers used s ...
of a given element is the element's
centralizer 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 ...
.Grillet (2007), p. 56/ref> Similarly, we can define a group action of $G$ on the set of all
subset 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 ...

s of $G,$ by writing $g \cdot S := gSg^,$ or on the set of the subgroups of $G.$

# Conjugacy class equation

If $G$ is a
finite 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 (mathematics), ...
, then for any group element $a,$ the elements in the conjugacy class of $a$ are in one-to-one correspondence with
coset 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). ...
s of the
centralizer 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 ...
$\operatorname_G\left(a\right).$ This can be seen by observing that any two elements $b$ and $c$ belonging to the same coset (and hence, $b = cz$ for some $z$ in the centralizer $\operatorname_G\left(a\right)$) give rise to the same element when conjugating $a$: $bab^ = cza(cz)^ = czaz^c^ = cazz^c^ = cac^.$ That can also be seen from the orbit-stabilizer theorem, when considering the group as acting on itself through conjugation, so that orbits are conjugacy classes and stabilizer subgroups are centralizers. The converse holds as well. Thus the number of elements in the conjugacy class of $a$ is 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 ...

## Example

Consider a finite $p$-group $G$ (that is, a group with order $p^n,$ where $p$ is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
and $n > 0$). We are going to prove that . Since the order of any conjugacy class of $G$ must divide the order of $G,$ it follows that each conjugacy class $H_i$ that is not in the center also has order some power of $p^,$ where $0 < k_i < n.$ But then the class equation requires that $, G, = p^n = , \operatorname(G), + \sum_i p^.$ From this we see that $p$ must divide $, \operatorname\left(G\right), ,$ so $, \operatorname\left(G\right), > 1.$ In particular, when $n = 2,$ then $G$ is an abelian group since any non-trivial group element is of order $p$ or $p^2.$ If some element $a$ of $G$ is of order $p^2,$ then $G$ is isomorphic to the cyclic group of order $p^2,$ hence abelian. On the other hand, if every non-trivial element in $G$ is of order $p,$ hence by the conclusion above $, \operatorname\left(G\right), > 1,$ then $, \operatorname\left(G\right), = p > 1$ or $p^2.$ We only need to consider the case when $, \operatorname\left(G\right), = p > 1,$ then there is an element $b$ of $G$ which is not in the center of $G.$ Note that $\operatorname_G\left(b\right)$ includes $b$ and the center which does not contain $b$ but at least $p$ elements. Hence the order of $\operatorname_G\left(b\right)$ is strictly larger than $p,$ therefore $\left, \operatorname_G\left(b\right)\ = p^2,$ therefore $b$ is an element of the center of $G,$ a contradiction. Hence $G$ is abelian and in fact isomorphic to the direct product of two cyclic groups each of order $p.$

# Conjugacy of subgroups and general subsets

More generally, given any
subset 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 ...

$S \subseteq G$ ($S$ not necessarily a subgroup), define a subset $T \subseteq G$ to be conjugate to $S$ if there exists some $g \in G$ such that $T = gSg^.$ Let $\operatorname\left(S\right)$ be the set of all subsets $T \subseteq G$ such that $T$ is conjugate to $S.$ A frequently used theorem is that, given any subset $S \subseteq G,$ 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 ...
of $\operatorname\left(S\right)$ (the
normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group (mathematics), group ''G'' is the set of elements C_G(S) of ''G'' such that each member g \in C_G(S) commutativity, commutes with ea ...
of $S$) in $G$ equals the order of $\operatorname\left(S\right)$: This follows since, if $g, h \in G,$ then $gSg^ = hSh^$ if and only if $g^h \in \operatorname\left(S\right),$ in other words, if and only if $g \text h$ are in the same
coset 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). ...
of $\operatorname\left(S\right).$ By using $S = \,$ this formula generalizes the one given earlier for the number of elements in a conjugacy class. The above is particularly useful when talking about subgroups of $G.$ 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 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 ...
, 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 In the mathematical 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 ...

of a
path-connected In topology 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) ...
topological space can be thought of as equivalence classes of
free loop In the mathematics, mathematical field of topology, a free loop is a variant of the mathematical notion of a Loop (topology), loop. Whereas a loop has a distinguished point on it, called a basepoint, a free loop lacks such a distinguished point. F ...
s under free homotopy.

# Conjugacy class and irreducible representations in finite group

In any
finite 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 (mathematics), ...
, the number of distinct (non-isomorphic)
irreducible representations 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 ...
over the complex numbers is precisely the number of conjugacy classes.

* * *

*