Editing Conjugacy class

{{short description|In group theory, equivalence class under the relation of conjugation}}
[[File:Dihedral-conjugacy-classes.svg|thumb|420px|Two [[Cayley graph]]s of [[dihedral group]]s with conjugacy classes distinguished by color.]]

In [[mathematics]], especially [[group theory]], two elements <math>a</math> and <math>b</math> of a [[Group (mathematics)|group]] are '''conjugate''' if there is an element <math>g</math> in the group such that <math>b = gag^{-1}.</math> This is an [[equivalence relation]] whose [[equivalence class]]es are called '''conjugacy classes'''.  In other words, each conjugacy class is closed under <math>b = gag^{-1}</math> for all elements <math>g</math> 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 group]]s is fundamental for the study of their structure.<ref name="dummit">{{cite book|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|title=Abstract Algebra|publisher=[[John Wiley & Sons]]|year=2004|edition=3rd|isbn=0-471-43334-9}}</ref><ref>{{cite book|last=Lang|first=Serge|author-link=Serge Lang|title=Algebra|publisher=[[Springer Science+Business Media|Springer]]|series=[[Graduate Texts in Mathematics]]|year=2002|isbn=0-387-95385-X}}</ref> For an [[abelian group]], each conjugacy class is a [[Set (mathematics)|set]] containing one element ([[Singleton (mathematics)|singleton set]]).

[[function (mathematics)|Function]]s that are constant for members of the same conjugacy class are called [[class function]]s.

==Definition<!--'Class number (group theory)' redirects here-->==

Let <math>G</math> be a group. Two elements <math>a, b \in G</math> are '''conjugate''' if there exists an element <math>g \in G</math> such that <math>gag^{-1} = b,</math> in which case <math>b</math> is called {{em|a conjugate}} of <math>a</math> and <math>a</math> is called a conjugate of <math>b.</math>

In the case of the [[general linear group]] <math>\operatorname{GL}(n)</math> of [[invertible matrices]], the conjugacy relation is called  [[matrix similarity]].

It can be easily shown that conjugacy is an equivalence relation and therefore partitions <math>G</math> into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes <math>\operatorname{Cl}(a)</math> and <math>\operatorname{Cl}(b)</math> are equal [[if and only if]] <math>a</math> and <math>b</math> are conjugate, and [[Disjoint sets|disjoint]] otherwise.) The equivalence class that contains the element <math>a \in G</math> is
<math display="block">\operatorname{Cl}(a) = \left\{ gag^{-1} : g \in G \right\}</math>
and is called the '''conjugacy class''' of <math>a.</math> The '''{{visible anchor|class number|Class number (group theory)}}'''<!--boldface per WP:R#PLA--> of <math>G</math> is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same [[Order (group theory)|order]].

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 [[Permutation#Cycle type|cycle type]].

==Examples==
The symmetric group [[Dihedral group of order 6|<math>S_3,</math>]] consisting of the 6 [[permutation]]s of three elements, has three conjugacy classes:

# No change <math>(abc \to abc)</math>.  The single member has order 1.
# [[Cyclic permutation#Transpositions|Transposing]] two <math>(abc \to acb, abc \to bac, abc \to cba)</math>.  The 3 members all have order 2.
# A [[cyclic permutation]] of all three <math>(abc \to bca, abc \to cab)</math>.  The 2 members both have order 3.

These three classes also correspond to the classification of the [[Isometry group|isometries]] of an [[equilateral triangle]].

[[File:Symmetric group S4; conjugacy table.svg|thumb|300px|Table showing <math>bab^{-1}</math> for all pairs <math>(a, b)</math> with <math>a, b \in S_4</math> <small>(compare [[:File:Symmetric group 4; permutation list.svg|numbered list]])</small>. Each row contains all elements of the conjugacy class {{nowrap|of <math>a,</math>}} and each column contains all elements of <math>S_4.</math>]]
The '''symmetric group [[v:Symmetric group S4|<math>S_4,</math>]]''' consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their description, [[Permutation#Cycle type|cycle type]], member order, and members:

# No change.  Cycle type = [1<sup>4</sup>].  Order = 1.  Members = { (1, 2, 3, 4) }.  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 = [1<sup>2</sup>2<sup>1</sup>].  Order = 2.  Members = { (1, 2, 4, 3), (1, 4, 3, 2), (1, 3, 2, 4), (4, 2, 3, 1), (3, 2, 1, 4), (2, 1, 3, 4) }).  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 = [1<sup>1</sup>3<sup>1</sup>].  Order = 3.  Members = { (1, 3, 4, 2), (1, 4, 2, 3), (3, 2, 4, 1), (4, 2, 1, 3), (4, 1, 3, 2), (2, 4, 3, 1), (3, 1, 2, 4), (2, 3, 1, 4) }).  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 = [4<sup>1</sup>].  Order = 4.  Members = { (2, 3, 4, 1), (2, 4, 1, 3), (3, 1, 4, 2), (3, 4, 2, 1), (4, 1, 2, 3), (4, 3, 1, 2) }).  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<sup>2</sup>].  Order = 2.  Members = { (2, 1, 4, 3), (4, 3, 2, 1), (3, 4, 1, 2) }).  The 3 rows containing this conjugacy class are shown with boldface entries in the adjacent table.

The [[Octahedral symmetry#The isometries of the cube|proper rotations of the cube]], which can be characterized by permutations of the body diagonals, are also described by conjugation in <math>S_4.</math>

In general, the number of conjugacy classes in the '''symmetric group <math>S_n</math>''' is equal to the number of [[integer partition]]s of <math>n.</math>  This is because each conjugacy class corresponds to exactly one partition of <math>\{ 1, 2, \ldots, n \}</math> into [[Cycle notation|cycles]], up to permutation of the elements of <math>\{ 1, 2, \ldots, n \}.</math>

In general, the '''[[Euclidean group]]''' can be studied by [[conjugation of isometries in Euclidean space]].

Example

Let G = <math>S_3</math>

<math>a = ( 2 3 )</math>

<math>x = ( 1 2 3 )</math>

<math>x^{-1} = ( 3 2 1 )</math>

Then <math>x a x^{-1}</math>

<math>= ( 1 2 3 ) ( 2 3 ) ( 3 2 1 ) = ( 3 1 )</math>

<math>= ( 3 1 )</math> is Conjugate of <math>( 2 3 )</math>

==Properties==

* The [[identity element]] is always the only element in its class, that is <math>\operatorname{Cl}(e) = \{ e \}.</math>
* If <math>G</math> is [[abelian group|abelian]] then <math>gag^{-1} = a</math> for all <math>a, g \in G</math>, i.e. <math>\operatorname{Cl}(a) = \{ a \}</math> for all <math>a \in G</math> (and the converse is also true: if all conjugacy classes are singletons then <math>G</math> is abelian).
* If two elements <math>a, b \in G</math> belong to the same conjugacy class (that is, if they are conjugate), then they have the same [[Order (group theory)|order]]. More generally, every statement about <math>a</math> can be translated into a statement about <math>b = gag^{-1},</math> because the map <math>\varphi(x) = gxg^{-1}</math> is an [[group isomorphism#Automorphisms|automorphism]] of <math>G</math> called an [[inner automorphism]]. See the next property for an example.
* If <math>a</math> and <math>b</math> are conjugate, then so are their powers <math>a^k</math> and <math>b^k.</math> (Proof: if <math>a = gbg^{-1}</math> then <math>a^k = \left(gbg^{-1}\right)\left(gbg^{-1}\right) \cdots \left(gbg^{-1}\right) = gb^kg^{-1}.</math>) Thus taking {{mvar|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 <math>a</math> is a power-up class of <math>a^k</math>).
* An element <math>a \in G</math> lies in the [[Center of a group|center]] <math>\operatorname{Z}(G)</math> of <math>G</math> if and only if its conjugacy class has only one element, <math>a</math> itself. More generally, if <math>\operatorname{C}_G(a)</math> denotes the {{em|[[centralizer]]}} of <math>a \in G,</math> i.e., the [[subgroup]] consisting of all elements <math>g</math> such that <math>ga = ag,</math> then the [[Index of a subgroup|index]] <math>\left[G : \operatorname{C}_G(a)\right]</math> is equal to the number of elements in the conjugacy class of <math>a</math> (by the [[orbit-stabilizer theorem]]).
* Take <math>\sigma \in S_n</math> and let <math>m_1, m_2, \ldots, m_s</math> be the distinct integers which appear as lengths of cycles in the cycle type of <math>\sigma</math> (including 1-cycles). Let <math>k_i</math> be the number of cycles of length <math>m_i</math> in <math>\sigma</math> for each <math>i = 1, 2, \ldots, s</math> (so that <math>\sum\limits_{i=1}^s k_i m_i = n</math>). Then the number of conjugates of <math>\sigma</math> is:<ref name="dummit" /><math display="block">\frac{n!}{\left(k_1!m_1^{k_1}\right) \left(k_2!m_2^{k_2}\right) \cdots \left(k_s!m_s^{k_s}\right)}.</math>

==Conjugacy as group action==<!-- This section is linked from [[Quaternions and spatial rotation]] -->

For any two elements <math>g, x \in G,</math> let
<math display="block">g \cdot x := gxg^{-1}.</math>
This defines a [[Group action (mathematics)|group action]] of <math>G</math> on <math>G.</math> The [[Group action (mathematics)#Orbits and stabilizers|orbits]] of this action are the conjugacy classes, and the [[Group action (mathematics)#Orbits and stabilizers|stabilizer]] of a given element is the element's [[centralizer]].<ref name="Grillet-2007-p56">Grillet (2007), [{{Google books|plainurl=y|id=LJtyhu8-xYwC|page=56|text=the orbits are the conjugacy classes}} p. 56]</ref>

Similarly, we can define a group action of <math>G</math> on the set of all [[subset]]s of <math>G,</math> by writing
<math display="block">g \cdot S := gSg^{-1},</math>
or on the set of the subgroups of <math>G.</math>

==Conjugacy class equation==

If <math>G</math> is a [[finite group]], then for any group element <math>a,</math> the elements in the conjugacy class of <math>a</math> are in one-to-one correspondence with [[coset]]s of the [[centralizer]] <math>\operatorname{C}_G(a).</math> This can be seen by observing that any two elements <math>b</math> and <math>c</math> belonging to the same coset (and hence, <math>b = cz</math> for some <math>z</math> in the centralizer <math>\operatorname{C}_G(a)</math>) give rise to the same element when conjugating <math>a</math>: 
<math display="block">bab^{-1} = cza(cz)^{-1} = czaz^{-1}c^{-1} = cazz^{-1}c^{-1} = cac^{-1}.</math>
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 <math>a</math> is the [[Index of a subgroup|index]] <math>\left[ G : \operatorname{C}_G(a)\right]</math> of the centralizer <math>\operatorname{C}_G(a)</math> in <math>G</math>; hence the size of each conjugacy class divides the order of the group.

Furthermore, if we choose a single representative element <math>x_i</math> from every conjugacy class, we infer from the disjointness of the conjugacy classes that 
<math display="block">|G| = \sum_i \left[ G : \operatorname{C}_G(x_i)\right],</math> 
where <math>\operatorname{C}_G(x_i)</math> is the centralizer of the element <math>x_i.</math> Observing that each element of the center <math>\operatorname{Z}(G)</math> forms a conjugacy class containing just itself gives rise to the '''class equation''':<ref>Grillet (2007), [{{Google books|plainurl=y|id=LJtyhu8-xYwC|page=57|text=The Class Equation}} p. 57]</ref>
<math display="block">|G| = |{\operatorname{Z}(G)}| + \sum_i \left[G : \operatorname{C}_G(x_i)\right],</math>
where the sum is over a representative element from each conjugacy class that is not in the center.

Knowledge of the divisors of the group order <math>|G|</math> can often be used to gain information about the order of the center or of the conjugacy classes.

===Example===

Consider a finite [[p-group|<math>p</math>-group]] <math>G</math> (that is, a group with order <math>p^n,</math> where <math>p</math> is a [[prime number]] and <math>n > 0</math>). We are going to prove that {{em|every finite <math>p</math>-group has a non-[[Trivial (mathematics)|trivial]] center}}.

Since the order of any conjugacy class of <math>G</math> must divide the order of <math>G,</math> it follows that each conjugacy class <math>H_i</math> that is not in the center also has order some power of <math>p^{k_i},</math> where <math>0 < k_i < n.</math> But then the class equation requires that <math display="inline">|G| = p^n = |{\operatorname{Z}(G)}| + \sum_i p^{k_i}.</math> From this we see that <math>p</math> must divide <math>|{\operatorname{Z}(G)}|,</math> so <math>|\operatorname{Z}(G)| > 1.</math>

In particular, when <math>n = 2,</math> then <math>G</math> is an abelian group since any non-trivial group element is of order <math>p</math> or <math>p^2.</math> If some element <math>a</math> of <math>G</math> is of order <math>p^2,</math> then <math>G</math> is isomorphic to the [[cyclic group]] of order <math>p^2,</math> hence abelian. On the other hand, if every non-trivial element in <math>G</math> is of order <math>p,</math> hence by the conclusion above <math>|\operatorname{Z}(G)| > 1,</math> then <math>|\operatorname{Z}(G)| = p > 1</math> or <math>p^2.</math> We only need to consider the case when <math>|\operatorname{Z}(G)| = p > 1,</math> then there is an element <math>b</math> of <math>G</math> which is not in the center of <math>G.</math> Note that <math>\operatorname{C}_G(b)</math> includes <math>b</math> and the center which does not contain <math>b</math> but at least <math>p</math> elements.  Hence the order of <math>\operatorname{C}_G(b)</math> is strictly larger than <math>p,</math> therefore <math>\left|\operatorname{C}_G(b)\right| = p^2,</math> therefore <math>b</math> is an element of the center of <math>G,</math> a contradiction. Hence <math>G</math> is abelian and in fact isomorphic to the direct product of two cyclic groups each of order <math>p.</math>

==Conjugacy of subgroups and general subsets==<!-- This section is linked from [[Symmetry group]] -->

More generally, given any [[subset]] <math>S \subseteq G</math> (<math>S</math> not necessarily a subgroup), define a subset <math>T \subseteq G</math> to be conjugate to <math>S</math> if there exists some <math>g \in G</math> such that <math>T = gSg^{-1}.</math> Let <math>\operatorname{Cl}(S)</math> be the set of all subsets <math>T \subseteq G</math> such that <math>T</math> is conjugate to <math>S.</math>

A frequently used theorem is that, given any subset <math>S \subseteq G,</math> the [[index of a subgroup|index]] of <math>\operatorname{N}(S)</math> (the [[normalizer]] of <math>S</math>) in <math>G</math> equals the cardinality of <math>\operatorname{Cl}(S)</math>:
<math display=block>|{\operatorname{Cl}(S)}| = [G : N(S)].</math>

This follows since, if <math>g, h \in G,</math> then <math>gSg^{-1} = hSh^{-1}</math> if and only if <math>g^{-1}h \in \operatorname{N}(S),</math> in other words, if and only if <math>g \text{ and } h</math> are in the same [[coset]] of <math>\operatorname{N}(S).</math>

By using <math>S = \{ a \},</math> 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 <math>G.</math> 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 [[Group isomorphism|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 loop]]s under free homotopy.

==Conjugacy class and irreducible representations in finite group==

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

==See also==
* {{annotated link|Topological conjugacy}}
* {{annotated link|FC-group}}
* {{annotated link|Conjugacy-closed subgroup}}

==Notes==
{{reflist}}

==References==
* {{cite book|last1=Grillet|first1=Pierre Antoine |title=Abstract algebra|edition=2|series=Graduate texts in mathematics|volume=242|year=2007|publisher=Springer|isbn=978-0-387-71567-4}}

==External links==
* {{springer|title=Conjugate elements|id=p/c025010}}

[[Category:Group theory]]