> [!NOTE] Theorem (Group Acts on Itself by Conjugation) > Let $G$ be a [[Groups|group]]. The [[Conjugate of Subgroup|conjugacy]] action $G\times G\to G, \quad \quad g*h=ghg^{-1}$is a [[Group action|action]] of $G$ on itself. ###### Proof that Group Acts on Itself by Conjugation Let $h\in G$. The conjugacy action fulfils the first group action axiom as $1_{G}h 1_{G}^{-1}=1_{G}h=h$. Let $g_{1},g_{2}\in G$. Then $\begin{align} (g_{1}g_{2})*h &= g_{1}g_{2} h (g_{1}g_{2})^{-1} \\ &= g_{1}g_{2} hg_{2}^{-1}g_{1}^{-1} &\text{inverse of group product} \\ &= g_{1} (g_{2}hg_{2}^{-1})g_{1}^{-1} \\ &= g_{1}(g_{2}*h) \end{align}$thus the conjugacy action fulfils the second group action axiom. # Application(s) Let $g\in G$. Then [[Orbit under Group Action|orbit]] of $g$ under this conjugation action is known as the conjugacy class of $g$, denoted $\text{C}_{G}(g)$, while the [[Stabilizer under Group Action|stabiliser]] of $g$ is known as the centraliser of $g$, denoted $\text{Cl}_{G}(g)$. The [[Orbit-Stabilizer Theorem|orbit-stabiliser theorem]] asserts that $\# G= \# \text{C}_{G}(g) \times\#\text{Cl}_{G}(g)$.