Conjugation Action - Knowlpedia

Let $G$ be a group. The conjugation action of $G$ on itself is the group action

g\cdot x := gxg^{-1}.

Remarks

Under this action, two elements lie in the same orbit exactly when they are conjugate, so the orbits are the conjugacy classes. The stabilizer of $x$ is its centralizer, and the kernel is the center.

More generally, $G$ acts on its subgroups by $g\cdot H := gHg^{-1}$ . The stabilizer of $H$ is its normalizer, and $H$ is normal if and only if every element of $G$ fixes it.

Examples

In $S_3$ , the conjugacy classes are $\{e\}$ , the three transpositions, and the two $3$ -cycles.
If $G$ is abelian, every conjugacy class is a singleton.