Conjugation Action

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

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 of the action is the center , consisting of elements that commute with all of $G$.

More generally, $G$ acts on its subgroups by $g\cdot H := gHg^{-1}$; the stabilizer of a subgroup $H$ in this action is its normalizer . A subgroup is normal iff it is fixed by every element under this action.

Examples:

• In $S_3$, the conjugacy classes are $\{e\}$, the three transpositions, and the two $3$-cycles.
• If $G$ is abelian, then $gxg^{-1}=x$ for all $g,x$, so every conjugacy class is a singleton.
• For the subgroup action, $H\le G$ is normal exactly when $gHg^{-1}=H$ for all $g\in G$.