Conjugacy class

In mathematics, especially group theory, two elements

In other words, each conjugacy class is closed under

Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties.

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

is the number of distinct (nonequivalent) conjugacy classes.

All elements belonging to the same conjugacy class have the same order.

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.

consisting of the 6 permutations of three elements, has three conjugacy classes: These three classes also correspond to the classification of the isometries of an equilateral triangle.

consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their description, cycle type, member order, and members: 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

[3] Similarly, we can define a group action of

are in one-to-one correspondence with cosets of 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.

Thus the number of elements in the conjugacy class of

; hence the size of each conjugacy class divides the order of the group.

Furthermore, if we choose a single representative element

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.

is isomorphic to the cyclic group of order

is abelian and in fact isomorphic to the direct product of two cyclic groups each of order

not necessarily a subgroup), define a subset

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

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.