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Encyclopedia > Conjugacy class
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In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. For abelian groups the concept is trivial, since each class is a set of one element (singleton set). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... A partition of U into 6 blocks: a Venn diagram representation. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a singleton is a set with exactly one element. ...

Contents


Definition

Suppose G is a group. Two elements a and b of G are called conjugate iff there exists an element g in G with ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

gag−1 = b.

(In linear algebra, for matrices this is called similarity). Several equivalence relations in mathematics are called similarity. ...


It can be readily 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 Cl(a) and Cl(b) are equal if and only if a and b are conjugate, and disjoint otherwise.) The equivalence class that contains the element a in G is In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, two sets are said to be disjoint if they have no element in common. ...

Cl(a) = {xG : there exists g in G such that x = gag−1}

and is called the conjugacy class of a.


Examples

The symmetric group S3, consisting of all 6 permutations of three elements, has three conjugacy classes: In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

  • no change (abc -> abc)
  • interchanging two (abc -> acb, abc -> bac, abc -> cba)
  • a cyclic permutation of all three (abc -> bca, abc -> cab)

The symmetric group S4, consisting of all 24 permutations of four elements, has five conjugacy classes:

  • no change
  • interchanging two
  • a cyclic permutation of three
  • a cyclic permutation of all four
  • interchanging two, and also the other two

See also the proper rotations of the cube, which can be characterized by permutations of the body diagonals. Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...


Properties

If G is abelian, then gag−1 = a for all a and g in G; so Cl(a) = {a} for all a in G; the concept is therefore not very useful in the abelian case. In mathematics, an abelian group is a commutative group, i. ...


If two elements a and b of G belong to the same conjugacy class (i.e. if they are conjugate), then they have the same order. More generally, every statement about a can be translated into a statement about b=gag−1, because the map φ(x) = gxg−1 is an automorphism of G. In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...


An element a of G lies in the center Z(G) of G if and only if its conjugacy class has only one element, a itself. More generally, if CG(a) denotes the centralizer of a in G, i.e. the subgroup consisting of all elements g such that ga = ag, then the index [G : CG(a)] is equal to the number of elements in the conjugacy class of a. In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if... In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...


For any character, the value of the character is the same for all members of a conjugacy class.


Conjugacy class equation

If G is a finite group, then the previous paragraphs, together with the Lagrange's theorem, imply that the number of elements in every conjugacy class divides the order of G. In mathematics, most commonly, Lagranges theorem states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. This can be shown using the concept of left cosets of H... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...


Furthermore, for any group G, we can define a representative set S = {xi} by picking one element from each conjugacy class of G that has more than one element. Then G is the disjoint union of Z(G) and the conjugacy classes Cl(xi) of the elements of S. One can then formulate the following important class equation:

|G| = |Z(G)| + ∑i [G : Hi]

where the sum extends over Hi = CG(xi) for each xi in S. Note that [G : Hi] is the number of elements in conjugacy class i, a proper divisor of |G| bigger than one. If the divisors of |G| are known, then this equation can often be used to gain information about the size of the center or of the conjugacy classes. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...


As an example of the usefulness of the class equation, consider a p-group G (that is, a group with order pn, where p is a prime number and n > 0). Since the order of any subgroup of G must divide the order of G, it follows that each Hi also has order some power of p( ki ). But then the class equation requires that |G| = pn = |Z(G)| + ∑i (p( ki )). From this we see that p must divide |Z(G)|, so |Z(G)| > 1, and therefore we have the result: every finite p-group has a non-trivial center. In mathematics, given a prime number p, a p-group is a group in which each element has a power of p as its order. ... Jump to: navigation, search In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... The word trivial has several common uses: Something that anyone can understand and explain to others — as opposed to something sublime, transcendental, etc. ...


Conjugacy of subgroups and general subsets

More generally, given any subset S of G (S not necessarily a subgroup), we define a subset T of G to be conjugate to S if and only if there exists some g in G such that T = gSg−1. We can define Cl(S) as the set of all subsets T of G such that T is conjugate to S. A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...


A frequently used theorem is that, given any subset S of G, the index of N(S) (the normalizer of S) in G equals the order of Cl(S): In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ...

|Cl(S)| = [G : N(S)]

This follows since, if g and h are in G, then gSg−1 = hSh−1 if and only if gh −1 is in N(S), in other words, if and only if g and h are in the same coset of N(S). In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...


Note that this formula generalizes the one given earlier for the number of elements in a conjugacy class (let S = {a}).


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 iff 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). In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...


Conjugacy as group action

If we define

g.x = gxg−1

for any two elements g and x in G, then we have a group action of G on G. The orbits of this action are the conjugacy classes, and the stabilizer of a given element is the element's centralizer. This article is about the mathematical concept. ...


Similarly, we can define a group action of G on the set of all subsets of G, by writing

g.S = gSg−1,

or on the set of the subgroups of G.


  Results from FactBites:
 
PlanetMath: conjugacy class formula (100 words)
The conjugacy classes of a group form a partition of its elements.
In a finite group, this means that the order of the group is the sum of the number of elements of the distinct conjugacy classes.
This is version 3 of conjugacy class formula, born on 2002-11-27, modified 2002-12-11.
Conjugacy class - Wikipedia, the free encyclopedia (887 words)
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.
For any character, the value of the character is the same for all members of a conjugacy class.
The orbits of this action are the conjugacy classes, and the stabilizer of a given element is the element's centralizer.
  More results at FactBites »

 
 

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