Conjugacy class

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.

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 g^-1ag = b. It can be readily shown that conjugacy is an equivalence relation and therefore partition 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 = g^-1ag}

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.

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 = g^-1Sg (the notation S ^x = x^-1Sx is often used). We can define Cl(S) as the set of all subsets T of G such that T is conjugate to S.

Properties

If G is abelian, then g^-1ag = a for all a and g in G; so Cl(a) = {a} for all a in G 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=g^-1ag, because the map φ(x) = g^-1xg is an automorphism of G. Similarly, if H and K are subgroups of G and H and K are conjugate, then H is isomorphic to K (note that the converse is not true; consider any isomorphic subgroups of an abelian group).

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 x and y are in G, then S ^x = S ^y if and only if xy ^-1 is in N(S), in other words, if and only if x and y are in the same coset of N(S).

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 (as can be seen by letting S = {a} in the equation above).

Conjugacy Class Equation

If G is a finite group, then the previous paragraphs, together with the Theorem of Lagrange, implies 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 equivalence class of G which has more than one element. S then has the property that 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. [G : H_i] is the number of elements in 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.