Action by conjugation

Example

Let $G$ be a group. The map $G \times G \to G$, $(g, x) \mapsto gxg^{-1}$, is an action of $G$ on the set $G$. Indeed, $exe^{-1} = x$ and $g(hxh^{-1})g^{-1} = (gh)x(gh)^{-1}$. The orbits of this action are the conjugacy classes of $G$. The stabiliser of $x$ is the centraliser $C_G(x)$. The fixed points are the elements of the centre $Z(G)$. For an abelian group, each element is a fixed point, and each orbit has one element.