Conjugacy class


In mathematics, especially group theory, two elements and of a group are conjugate if there is an element in the group such that This is an equivalence relation whose equivalence classes are called conjugacy classes.

Two Cayley graphs of dihedral groups with conjugacy classes distinguished by color.

Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study 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.


Let   be a group. Two elements   are conjugate if there exists an element   such that   in which case   is called a conjugate of   and   is called a conjugate of  

In the case of the general linear group   of invertible matrices, the conjugacy relation is called matrix similarity.

It can be easily shown that conjugacy is an equivalence relation and therefore partitions   into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes   and   are equal if and only if   and   are conjugate, and disjoint otherwise.) The equivalence class that contains the element   is

and is called the conjugacy class of   The class number of   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   consisting of the 6 permutations of three elements, has three conjugacy classes:

  1. no change  
  2. transposing two  
  3. a cyclic permutation of all three  

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

Table showing   for all pairs   with   (compare numbered list). Each row contains all elements of the conjugacy class of   and each column contains all elements of  

The symmetric group   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  

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

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  
  • If   is abelian then   for all  , i.e.   for all   (and the converse is also true: if all conjugacy classes are singletons then   is abelian).
  • If two elements   belong to the same conjugacy class (that is, if they are conjugate), then they have the same order. More generally, every statement about   can be translated into a statement about   because the map   is an automorphism of   See the next property for an example.
  • If   and   are conjugate, then so are their powers   and   (Proof: if   then  ) Thus taking  th 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   is a power-up class of  ).
  • An element   lies in the center   of   if and only if its conjugacy class has only one element,   itself. More generally, if   denotes the centralizer of   i.e., the subgroup consisting of all elements   such that   then the index   is equal to the number of elements in the conjugacy class of   (by the orbit-stabilizer theorem).
  • Take   and let   be the distinct integers which appear as lengths of cycles in the cycle type of   (including 1-cycles). Let   be the number of cycles of length   in   for each   (so that  ). Then the number of conjugates of   is:[1]

Conjugacy as group actionEdit

For any two elements   let

This defines a group action of   on   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   on the set of all subsets of   by writing

or on the set of the subgroups of  

Conjugacy class equationEdit

If   is a finite group, then for any group element   the elements in the conjugacy class of   are in one-to-one correspondence with cosets of the centralizer   This can be seen by observing that any two elements   and   belonging to the same coset (and hence,   for some   in the centralizer  ) give rise to the same element when conjugating  :

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   is the index   of the centralizer   in  ; hence the size of each conjugacy class divides the order of the group.

Furthermore, if we choose a single representative element   from every conjugacy class, we infer from the disjointness of the conjugacy classes that

where   is the centralizer of the element   Observing that each element of the center   forms a conjugacy class containing just itself gives rise to the class equation:[4]
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   can often be used to gain information about the order of the center or of the conjugacy classes.


Consider a finite  -group   (that is, a group with order   where   is a prime number and  ). We are going to prove that every finite  -group has a non-trivial center.

Since the order of any conjugacy class of   must divide the order of   it follows that each conjugacy class   that is not in the center also has order some power of   where   But then the class equation requires that   From this we see that   must divide   so  

In particular, when   then   is an abelian group since any non-trivial group element is of order   or   If some element   of   is of order   then   is isomorphic to the cyclic group of order   hence abelian. On the other hand, if every non-trivial element in   is of order   hence by the conclusion above   then   or   We only need to consider the case when   then there is an element   of   which is not in the center of   Note that   includes   and the center which does not contain   but at least   elements. Hence the order of   is strictly larger than   therefore   therefore   is an element of the center of   a contradiction. Hence   is abelian and in fact isomorphic to the direct product of two cyclic groups each of order  

Conjugacy of subgroups and general subsetsEdit

More generally, given any subset   (  not necessarily a subgroup), define a subset   to be conjugate to   if there exists some   such that   Let   be the set of all subsets   such that   is conjugate to  

A frequently used theorem is that, given any subset   the index of   (the normalizer of  ) in   equals the order of  :


This follows since, if   then   if and only if   in other words, if and only if   are in the same coset of  

By using   this formula generalizes the one given earlier for the number of elements in a conjugacy class.

The above is particularly useful when talking about subgroups of   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 interpretationEdit

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 groupEdit

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

See alsoEdit


  1. ^ a b Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
  2. ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
  3. ^ Grillet (2007), p. 56
  4. ^ Grillet (2007), p. 57


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

External linksEdit