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).

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

gag−1 = b.

(In linear algebra, for matrices this is 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

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:

  • 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.

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.

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.

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.

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.

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.

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.

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 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):

|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).

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).

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.

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.