Conjugacy class

Definition

Let ${\displaystyle G}$ be a group. Two elements ${\displaystyle a,b\in G}$ are conjugate if there exists an element ${\displaystyle g\in G}$ such that ${\displaystyle ga{g}^{-1}=b,}$ in which case ${\displaystyle b}$ is called a conjugate of ${\displaystyle a}$ and ${\displaystyle a}$ is called a conjugate of ${\displaystyle b.}$

In the case of the general linear group ${\displaystyle \mathrm{GL}(n)}$ of invertible matrices, the conjugacy relation is called matrix similarity.

It can be easily shown that conjugacy is an equivalence relation and therefore partitions ${\displaystyle G}$ into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes ${\displaystyle \mathrm{Cl}(a)}$ and ${\displaystyle \mathrm{Cl}(b)}$ are equal if and only if ${\displaystyle a}$ and ${\displaystyle b}$ are conjugate, and disjoint otherwise.) The equivalence class that contains the element ${\displaystyle a\in G}$ is $${\displaystyle \mathrm{Cl}(a)=\left\{ga{g}^{-1}:g\in G\right\}}$$ and is called the conjugacy class of ${\displaystyle a.}$ The class number of ${\displaystyle G}$ 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 with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity which has order 1. In some cases, conjugacy classes can be described in a uniform way; for example, in the symmetric group they can be described by cycle type.

Properties

The identity element is always the only element in its class, that is ${\displaystyle \mathrm{Cl}(e)=\{e\}.}$ More generally, an element ${\displaystyle a\in G}$ lies in the center ${\displaystyle \mathrm{Z}(G)}$ of ${\displaystyle G}$ if and only if its conjugacy class has only one element, ${\displaystyle a}$ itself. This follows because if ${\displaystyle a\in \mathrm{Z}(G)}$ then ${\displaystyle ga{g}^{-1}=a}$ for all ${\displaystyle g\in G.}$ Hence if ${\displaystyle G}$ is abelian, ${\displaystyle \mathrm{Cl}(a)=\{a\}}$ for all ${\displaystyle a\in G}$ (and the converse is also true: if all conjugacy classes are singletons then ${\displaystyle G}$ is abelian).

If two elements ${\displaystyle a,b\in G}$ belong to the same conjugacy class (that is, if they are conjugate), then they have the same order. More generally, every statement about ${\displaystyle a}$ can be translated into a statement about ${\displaystyle b=ga{g}^{-1},}$ because the map ${\displaystyle \phi (x)=gx{g}^{-1},}$ as noted previously, is an automorphism of ${\displaystyle G}$.

As an example, if ${\displaystyle a}$ and ${\displaystyle b}$ are conjugate, then so are their powers ${\displaystyle a^k}$ and ${\displaystyle b^k.}$ (Proof: if ${\displaystyle a=gb{g}^{-1}}$ then ${\displaystyle a^k=\left(gb{g}^{-1}\right)\left(gb{g}^{-1}\right)\cdots \left(gb{g}^{-1}\right)=g{b}^k{g}^{-1}.}$) Thus taking the kth power 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 ${\displaystyle a}$ is a power-up class of ${\displaystyle a^k}$).

An inner automorphism on ${\displaystyle G}$ given by ${\displaystyle a\mapsto ga{g}^{-1}}$ is the identity automorphism if and only if ${\displaystyle g\in \mathrm{Z}(G)}$. This implies that the quotient group ${\displaystyle G/\mathrm{Z}(G)}$ is isomorphic to the group of inner automorphisms of ${\displaystyle G.}$

Examples

The symmetric group ${\displaystyle S_3,}$ consisting of the 6 permutations of three elements, has three conjugacy classes:

1. No change: ${\displaystyle (abc\to abc)}$
2. Transposing two: ${\displaystyle (abc\to acb,abc\to bac,abc\to cba)}$
3. A cyclic permutation of all three: ${\displaystyle (abc\to bca,abc\to cab)}$

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

For example, in the third class above (corresponding to rotations of the equilateral triangle by 120 and 240 degrees, respectively), a possible choice for g would be the transposition ${\displaystyle abc\to acb}$ from the second class (corresponding to the flipping of the triangle around its axis going through a, which is its own inverse): flipping a triangle vertically, then rotating it by 120 degrees, then flipping it again (${\displaystyle abc\to acb\to bac\to bca}$) is equivalent to rotating the unflipped triangle by 270 degrees (${\displaystyle abc\to bca}$).

The symmetric group ${\displaystyle S_4,}$ consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their members using cycle notation:

1. No change: ${\displaystyle \{(1)\}}$
2. Interchanging two: ${\displaystyle \{(12),(13),(14),(23),(24),(34)\}}$
3. A cyclic permutation of three: ${\displaystyle \{(123),(124),(132),(134),(142),(143),(234),(243)\}}$
4. A cyclic permutation of all four: ${\displaystyle \{(1234),(1243),(1324),(1342),(1423),(1432)\}}$
5. Interchanging two, and also the other two: ${\displaystyle \{(12)(34),(13)(24),(14)(23)\}}$

In general, the number of conjugacy classes in the symmetric group ${\displaystyle S_n}$ is equal to the number of integer partitions of ${\displaystyle n.}$ This is because each conjugacy class corresponds to exactly one partition of ${\displaystyle \{1,2,\dots ,n\}}$ into cycles, up to permutation of the elements of ${\displaystyle \{1,2,\dots ,n\}.}$ The size of ${\displaystyle \mathrm{Cl}(\sigma ),}$ for ${\displaystyle \sigma \in S_m,}$ can be computed from the cycle lengths. Let ${\displaystyle m_1,m_2,\dots ,m_s}$ be the distinct integers which appear as lengths of cycles in the cycle type of ${\displaystyle \sigma }$ (including 1-cycles) and let ${\displaystyle k_i}$ be the number of cycles of length ${\displaystyle m_i}$ for each ${\displaystyle i=1,2,\dots ,s}$ (so that $\sum _{i=1}^s{k}_i{m}_i=n$). Then the number of elements in ${\displaystyle \mathrm{Cl}(\sigma )}$ is$${\displaystyle \frac{n!}{\left(k_1!m_1^{k_1}\right)\left(k_2!m_2^{k_2}\right)\cdots \left(k_s!m_s^{k_s}\right)}.}$$

The dihedral group ${\displaystyle D_5}$ consisting of symmetries of a pentagon, has four conjugacy classes:

1. The identity element: ${\displaystyle \{1\}}$
2. Two conjugacy classes of size 2: ${\displaystyle \{r,r^4\},\{r^2,r^3\}}$
3. All the reflections: ${\displaystyle \{s,rs,r^{2}s,r^{3}s,r^{4}s\}}$

Conjugation as a group action, centralizers, and the class equation

For any two elements ${\displaystyle g,x\in G,}$ let $${\displaystyle g\cdot x:=gx{g}^{-1}.}$$ This defines a group action of ${\displaystyle G}$ on ${\displaystyle G.}$ The orbits of this action are the conjugacy classes. Let ${\displaystyle {\mathrm{C}}_G(a)}$ denote the centralizer of ${\displaystyle a\in G,}$ i.e., the subgroup consisting of all elements ${\displaystyle b\in G}$ such that ${\displaystyle ba=ab.}$ Then the stabilizer of a given element ${\displaystyle x\in G}$ is ${\displaystyle {\mathrm{C}}_G(x).}$ Moreover, the set of elements fixed by ${\displaystyle g\in G}$ under conjugation is ${\displaystyle {\mathrm{C}}_G(g).}$

Conjugacy class equation

For any element ${\displaystyle a}$ of a group ${\displaystyle G,}$ the elements of the conjugacy class of ${\displaystyle a}$ are in one-to-one correspondence with cosets of the centralizer ${\displaystyle {\mathrm{C}}_G(a).}$ This can be seen by observing that any two elements ${\displaystyle b}$ and ${\displaystyle c}$ belong to the same coset of ${\displaystyle {\mathrm{C}}_G(a),}$ meaning ${\displaystyle b=cz}$ for some ${\displaystyle z\in {\mathrm{C}}_G(a),}$ if and only if they give rise to the same element when conjugating ${\displaystyle a}$: $${\displaystyle ba{b}^{-1}=ca{c}^{-1}⟺\left(c^{-1}b\right)a=a\left(c^{-1}b\right)⟺z=c^{-1}b\in {\mathrm{C}}_G(a).}$$ This is a special case of the orbit-stabilizer theorem, keeping in mind that conjugacy classes are orbits and centralizers are stabilizers under the action of the group on itself through conjugation.

Thus if ${\displaystyle G}$ is a finite group, the number of elements in the conjugacy class of ${\displaystyle a}$ is the index ${\displaystyle \left[G:{\mathrm{C}}_G(a)\right]}$ of the centralizer ${\displaystyle {\mathrm{C}}_G(a)}$ in ${\displaystyle G}$; hence the size of each conjugacy class divides the order of the group.

Furthermore, if we choose a single representative element ${\displaystyle x_i}$ from every conjugacy class, we infer from the disjointness of the conjugacy classes that $${\displaystyle |G|=\sum _i\left[G:{\mathrm{C}}_G(x_i)\right].}$$ Observing that each element of the center ${\displaystyle \mathrm{Z}(G)}$ forms a conjugacy class containing just itself gives rise to the class equation: $${\displaystyle |G|=|\mathrm{Z}(G)|+\sum _i\left[G:{\mathrm{C}}_G(x_i)\right],}$$ 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 ${\displaystyle |G|}$ can often be used to gain information about the order of the center or of the conjugacy classes.

Example

Consider a finite ${\displaystyle p}$-group ${\displaystyle G}$ (that is, a group with order ${\displaystyle p^n,}$ where ${\displaystyle p}$ is a prime number and ${\displaystyle n>0}$). We are going to prove that every finite ${\displaystyle p}$-group has a center of size greater than 1.

Since the order of any conjugacy class of ${\displaystyle G}$ must divide the order of ${\displaystyle G,}$ it follows that each conjugacy class ${\displaystyle H_i}$ that is not in the center also has order some power of ${\displaystyle p^{k_i},}$ where ${\displaystyle 1\le k_i\le n-1.}$ But then the class equation requires that $|G|=p^n=|\mathrm{Z}(G)|+\sum _i{p}^{k_i}.$ From this we see that ${\displaystyle p}$ must divide ${\displaystyle |\mathrm{Z}(G)|,}$ so ${\displaystyle |\mathrm{Z}(G)|>1.}$

In particular, when ${\displaystyle n=2}$ we can further show that ${\displaystyle G}$ is abelian. From the foregoing, ${\displaystyle |\mathrm{Z}(G)|,}$ equals either ${\displaystyle p}$ or ${\displaystyle p^2}$ and, if ${\displaystyle G}$ were nonabelian, would have to equal ${\displaystyle p.}$ Furthermore, there would have to be an element ${\displaystyle b}$ not in ${\displaystyle \mathrm{Z}(G)}$. Its centralizer subgroup ${\displaystyle {\mathrm{C}}_G(b)}$ would, however, have to include both ${\displaystyle b}$ and all the elements of ${\displaystyle \mathrm{Z}(G),}$ implying ${\displaystyle |{\mathrm{C}}_G(b)|=p^2.}$ This contradicts ${\displaystyle b\notin \mathrm{Z}(G).}$ Hence ${\displaystyle G}$ is abelian and is, in fact, isomorphic either to a cyclic group of order ${\displaystyle p^2}$ or to the direct product of two cyclic groups of order ${\displaystyle p.}$

Average centralizer

By Burnside's lemma, the number of conjugacy classes of a finite group ${\displaystyle G}$ is equal to ${\displaystyle \frac{1}{|G|}\sum _g|C_G(g)|}$, the average size of the sets fixed by the elements of ${\displaystyle G}$ acting by conjugation, that is, the average size of the centralizers of elements of ${\displaystyle G.}$

Conjugacy of subgroups and general subsets

More generally, given any subset ${\displaystyle S\subseteq G}$ (${\displaystyle S}$ not necessarily a subgroup), define a subset ${\displaystyle T\subseteq G}$ to be conjugate to ${\displaystyle S}$ if there exists some ${\displaystyle g\in G}$ such that ${\displaystyle T=gS{g}^{-1}.}$ Let ${\displaystyle \mathrm{Cl}(S)}$ be the set of all subsets ${\displaystyle T\subseteq G}$ such that ${\displaystyle T}$ is conjugate to ${\displaystyle S.}$

Define a group action of ${\displaystyle G}$ on the set of all subsets of ${\displaystyle G,}$ by writing $${\displaystyle g\cdot S:=gS{g}^{-1}.}$$ The result for the number of elements in the conjugacy class of an element generalizes to the case of subsets using an argument analogous to the one used previously: there is a one-to-one correspondence between elements of the conjugacy class of ${\displaystyle S}$ and cosets of ${\displaystyle \mathrm{N}(S),}$ the normalizer of ${\displaystyle S,}$ since, if ${\displaystyle g,h\in G,}$ then ${\displaystyle gS{g}^{-1}=hS{h}^{-1}}$ if and only if ${\displaystyle g^{-1}h\in \mathrm{N}(S),}$ in other words, if and only if ${\displaystyle g\text{ and }h}$ are in the same coset of ${\displaystyle \mathrm{N}(S).}$ As a result, the index of ${\displaystyle \mathrm{N}(S)}$ in ${\displaystyle G}$ equals the cardinality of ${\displaystyle \mathrm{Cl}(S)}$: $${\displaystyle |\mathrm{Cl}(S)|=[G:N(S)].}$$ By using ${\displaystyle S=\{a\},}$ this formula specializes to the one given earlier for the number of elements in a conjugacy class since the normalizer of a singleton equals its centralizer.

The above is particularly useful when talking about subgroups of ${\displaystyle G.}$ 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.

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.

Conjugacy class and irreducible representations in finite group

In any finite group, the number of nonisomorphic irreducible representations over the complex numbers is precisely the number of conjugacy classes.