conjugacy class

conjugacy class (plural conjugacy classes)

1. (algebra) A subset of a group which is an equivalence class in the quotient set of the group divided by conjugation as equivalence relation.

   For a given element ${\displaystyle x}$ of some group ${\displaystyle G}$, its conjugacy class ${\displaystyle x^{G}}$ is ${\displaystyle \{y\in G|\exists g\in G:g^{-1}xg=y\}}$.

   If a group acts on itself through conjugation, then its orbits are its conjugacy classes and its stabilizer subgroups are its centralizers.