Conjugacy class

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Conjugacy class

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. In all abelian groups every conjugacy class is a set containing one element (singleton set).

Functions that are constant for members of the same conjugacy class are called class functions.

Definition

Suppose G is a group. Two elements a and b of G are called conjugate if there exists an element g in G with

gag−1 = b.

(In linear algebra, this is referred to as similarity of matrices.)

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) = { gag−1: gG }

and is called the conjugacy class of a. The class number of G is the number of distinct (nonequivalent) conjugacy classes.

Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class of order 6 elements", and "6B" would be a different conjugacy class of order 6 elements; the conjugacy class 1A is the conjugacy class of the identity. In some cases, conjugacy classes can be described in a uniform way – for example, in the symmetric group they can be described by cycle structure.

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, listed with their cycle structures and orders:

• (1)4: no change (1 element)
• (2): interchanging two (6 elements)
• (3): a cyclic permutation of three (8 elements)
• (4): a cyclic permutation of all four (6 elements)
• (2)(2): interchanging two, and also the other two (3 elements)

In general, the number of conjugacy classes in the symmetric group Sn is equal to the number of integer partitions of n. This is because each conjugacy class corresponds to exactly one partition of {1, 2, ..., n} into cycles, up to permutation of the elements of {1, 2, ..., n}.

See also the proper rotations of the cube, which can be characterized by permutations of the body diagonals.

Properties

• The identity element is always in its own class, that is Cl(e) = {e}
• 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. The failure of this thus gives us an idea in what degree the group is nonabelian.
• 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 (by the orbit-stabilizer theorem).
• If a and b are conjugate, then so are powers of them, ak and bk – thus taking kth powers gives a map on conjugacy classes, and one may speak of which conjugacy classes a given conjugacy class "powers up" into. 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), while the cube is an element of type (2), so the class (3)(2) powers up into the classes (3) and (2).

Conjugacy class equation

If G is a finite group, then for any group element a, the elements in the conjugacy class of a are in one-to-one correspondence with cosets of the centralizer CG(a). This can be seen by observing that any two elements b and c belonging to the same coset (and hence, b=cz for some z in the centralizer CG(a)) give rise to the same element when conjugating a: bab−1=cza(cz)−1=czaz-1c-1=czz-1ac-1=cac−1.

Thus the number of elements in the conjugacy class of a is the index [G:CG(a)] of the centralizer CG(a) in G. Lagrange's theorem then implies that the size of each conjugacy class is a divisor of the order of the group.

Furthermore, if we choose a single representative element xi from every conjugacy class, we infer from the disjointedness of the conjugacy classes that |G| = ∑i [G : CG(xi)], where CG(xi) is the centralizer of the element xi. Observing that each element of the center Z(G) forms a conjugacy class containing just itself gives rise to the following important class equation:

|G| = |Z(G)| + ∑i [G : CG(xi)]

where the second sum is over a representative element from each conjugacy class that is not in the center.

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

Example

Consider a finite p-group G (that is, a group with order pn, where p is a prime number and n > 0). We are going to prove that: every finite p-group has a non-trivial center.

Since the order of any conjugacy class of G must divide the order of G, it follows that each conjugacy class Hi also has order some power of p(ki), where 0 < ki < n. 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.

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 g−1h 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 if and only if 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.

Geometric interpretation

Conjugacy classes in the fundamental group of a path-connected topological space can be thought of as equivalence classes of free loops under free homotopy.

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