Conjugacy class

In mathematics, 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 reveals many important features of a group's structure.

Contents

1 Definition 2 Properties 3 Conjugacy class equation 4 Conjugacy of subgroups and general subsets 5 Conjugacy as group action

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. It can be readily shown that conjugacy is an equivalence relation and therefore partitions G into equivalence classes. The equivalence class that contains the element a in G is

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

and is called the conjugacy class of a. Every element of the group belongs to precisely one conjugacy class. The classes Cl(a) and Cl(b) are equal if and only if a and b are conjugate, and disjoint otherwise.

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 C_G(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 : C_G(a)] is equal to the number of elements in the conjugacy class of a.

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 = {x_i} 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(x_i) of the elements of S. One can then formulate the following important class equation:

|G| = |Z(G)| + ∑_i [G : H_i]

where the sum extends over H_i = C_G(x_i) for each x_i in S. Note that [G : H_i] 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 group G 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 H_i also has order some power of p^{( k_i )}. But then the class equation requires that |G| = pⁿ = |Z(G)| + ∑_i (p^{( k_i )}). 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 (consider any two isomorphic subgroups of an abelian group).

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.