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.^[1][2] For an abelian group, each 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

Let G be a group. Two elements a and b of G are conjugate, if there exists an element g in G such that gag⁻¹ = b. One says also that b is a conjugate of a and that a is a conjugate of b .

In the case of the group GL(n) of invertible matrices, the conjugacy relation is called matrix similarity.

It can be easily 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) = { b ∈ G | there exists g ∈ G with b = gag⁻¹ }

and is called the conjugacy class of a. The class number of G is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same order.

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)
• transposing two (abc → acb, abc → bac, abc → cba)
• a cyclic permutation of all three (abc → bca, abc → cab)

These three classes also correspond to the classification of the isometries of an equilateral triangle.

Table showing bab⁻¹ for all pairs (a, b) with a, b ∈ S₄ (compare numbered list)     Each row contains all elements of the conjugacy class of a, and each column contains all elements of S₄.

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: { (1, 2, 3, 4) } )

(2)

Interchanging two (6 elements: { (1, 2, 4, 3), (1, 4, 3, 2), (1, 3, 2, 4), (4, 2, 3, 1), (3, 2, 1, 4), (2, 1, 3, 4) })

(3)

A cyclic permutation of three (8 elements: { (1, 3, 4, 2), (1, 4, 2, 3), (3, 2, 4, 1), (4, 2, 1, 3), (4, 1, 3, 2), (2, 4, 3, 1), (3, 1, 2, 4), (2, 3, 1, 4) } )

(4)

A cyclic permutation of all four (6 elements: { (2, 3, 4, 1), (2, 4, 1, 3), (3, 1, 4, 2), (3, 4, 2, 1), (4, 1, 2, 3), (4, 3, 1, 2) } )

(2)(2)

Interchanging two, and also the other two (3 elements: { (2, 1, 4, 3), (4, 3, 2, 1), (3, 4, 1, 2) } )

The proper rotations of the cube, which can be characterized by permutations of the body diagonals, are also described by conjugation in S₄ .

In general, the number of conjugacy classes in the symmetric group S_n 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}.

In general, the Euclidean group can be studied by conjugation of isometries in Euclidean space.

Properties

• The identity element is always the only element in its 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.
• 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 (by the orbit-stabilizer theorem).
• If a and b are conjugate, then so are their powers a^k and b^k. (Proof: if a = gbg⁻¹, then a^k = (gbg⁻¹)(gbg⁻¹) … (gbg⁻¹) = gb^kg⁻¹.) Thus taking kth powers gives a map on conjugacy classes, and one may consider which conjugacy classes are in its preimage. 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), therefore one of the power-up classes of (3) is the class (3)(2); the class (6) is another.

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 C_G(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 C_G(a) ) give rise to the same element when conjugating a: bab⁻¹ = cza(cz)⁻¹ = czaz⁻¹c⁻¹ = czz⁻¹ac⁻¹ = cac⁻¹. The converse holds as well.

Thus the number of elements in the conjugacy class of a is the index [G : C_G(a)] of the centralizer C_G(a) in G ; hence the size of each conjugacy class divides the order of the group.

Furthermore, if we choose a single representative element x_i from every conjugacy class, we infer from the disjointness of the conjugacy classes that |G| = ∑_i [G : C_G(x_i)], where C_G(x_i) is the centralizer of the element x_i. Observing that each element of the center Z(G) forms a conjugacy class containing just itself gives rise to the class equation:^[3]

|G| = |Z(G)| + ∑_i [G : C_G(x_i)]

where the 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 H_i that is not in the center also has order some power of p^k_i, where 0 < k_i < n. 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 .

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 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.^[4]

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.

See also

• Topological conjugacy
• FC-group
• Conjugacy-closed subgroup

References

• Grillet, Pierre Antoine (2007). Abstract algebra. Graduate texts in mathematics. 242 (2 ed.). Springer. ISBN 978-0-387-71567-4.