Conjugacy class

In mathematics, especially group theory, two elements ${\displaystyle a}$ and ${\displaystyle b}$ of a group are conjugate if there is an element ${\displaystyle g}$ in the group such that ${\displaystyle b=gag^{-1}.}$ This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under ${\displaystyle b=gag^{-1}}$ for all elements ${\displaystyle g}$ in the group.

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.