# Conjugacy class

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.

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:

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

The **symmetric group ** consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their cycle structures and orders:

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

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.