Defintion

The orbits under these action are the Conjugacy classes.

The stabilizer is the Centralizer

The Orbit-stabilizer theorem for the conjugation action becomes the Class equation

Def. Let $H \subset G$ Subgroup. For every $$g \in G$, we can define the Conjugate subgroup $gHg^{-1} = \{ ghg^{-1}| h \in H\}$.

The Stabilizer subgroup for the conjugation action is called the Normalizer subgroup