Conjugation action of a Lie group

Let $G$ be a Lie group.

Definition. The conjugation action is the map

G\times G \to G,\quad (g,h)\mapsto ghg^{-1}.

This is a smooth action of $G$ on the manifold $G$ .

Remarks

Orbits and stabilizers.

The orbit of $h\in G$ is its conjugacy class, an example of an orbit of a Lie group action.
The stabilizer of $h$ is its centralizer $C_G(h)=\{g\in G:gh=hg\}$ , a closed subgroup; compare stabilizers and the closed subgroup theorem.
The kernel of the action is the center $Z(G)$ .

Infinitesimal picture. Differentiating conjugation at the identity yields the adjoint action $\mathrm{Ad}:G\to \mathrm{Aut}(\mathfrak{g})$ , so conjugation is the global geometric source of the adjoint representation.