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 the maps ${\displaystyle a\mapsto gag^{-1},}$ with ${\displaystyle g}$ an element of 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. 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.

The notion of a conjugate may be extended from elements to subsets: subsets ${\displaystyle S}$ and ${\displaystyle T}$ of ${\displaystyle G}$ are conjugate if there is an element ${\displaystyle g\in G}$ such that ${\displaystyle T=gSg^{-1},}$ where ${\displaystyle gSg^{-1}=\left\{gsg^{-1}:s\in S\right\}.}$ The conjugacy class of ${\displaystyle S}$ is the set of all subsets of ${\displaystyle G}$ conjugate to ${\displaystyle S.}$ A normal subgroup is defined by the property that its conjugacy class contains a single member, namely itself. Normal subgroups play a key role in the study of quotient groups and group homomorphisms.