Conjugacy class

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) = { gag⁻¹ | g ∈ G }

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.

The symmetric group S₃, consisting of the 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 the 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) }). The single row containing this conjugacy class is shown as a row of black circles in the adjacent table.

(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) }). The 6 rows containing this conjugacy class are highlighted in green in the adjacent table.

(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) }). The 8 rows containing this conjugacy class are shown with normal print (no boldface or color highlighting) in the adjacent table.

(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) }). The 6 rows containing this conjugacy class are highlighted in orange in the adjacent table.

(2)(2)  interchanging two, and also the other two (3 elements: { (2, 1, 4, 3), (4, 3, 2, 1), (3, 4, 1, 2) }). The 3 rows containing this conjugacy class are shown with boldface entries in the adjacent table.

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.

• 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. See the next property for an example.
• 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) (where a is a power-up class of a^k).
• 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).
• Take ${\displaystyle \sigma \in S_{n}}$ and let ${\displaystyle m_{1},m_{2},...,m_{s}}$ be the distinct integers which appear as lengths of cycles in the cycle type of ${\displaystyle \sigma }$ (including 1-cycles). Let ${\displaystyle k_{i}}$ be the number of cycles of length ${\displaystyle m_{i}}$ in ${\displaystyle \sigma }$ for each ${\displaystyle i=1,2,...,s}$ (so that ${\displaystyle \sum \limits _{i=1}^{s}k_{i}m_{i}=n}$). Then the number of conjugates of ${\displaystyle \sigma }$ is:[1]

${\displaystyle {\frac {n!}{(k_{1}!m_{1}^{k_{1}})(k_{2}!m_{2}^{k_{2}})...(k_{s}!m_{s}^{k_{s}})}}}$

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.[3]

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.

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 = cazz⁻¹c⁻¹ = cac⁻¹. That can also be seen from the orbit-stabilizer theorem, when considering the group as acting on itself through conjugation, so that orbits are conjugacy classes and stabilizer subgroups are centralizers. 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:[4]

|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.

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):

${\displaystyle |\operatorname {Cl} (S)|=[G:N(S)]}$

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

In any finite group, the number of distinct (non-isomorphic) irreducible representations over the complex numbers is precisely the number of conjugacy classes.

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

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

• "Conjugate elements", Encyclopedia of Mathematics, EMS Press, 2001 [1994]