State the class equation and derive it from orbit-stabilizer.
Solution
Let a finite group \(G\) act on itself by conjugation, \[ g\cdot x=gxg^{-1}. \] The orbit of \(x\) is its conjugacy class, and its stabilizer is its centralizer \[ C_G(x)=\{g\in G:gx=xg\}. \] By orbit-stabilizer, \[ |\operatorname{Cl}_G(x)|=[G:C_G(x)]. \]
The conjugacy classes partition \(G\). The classes of size \(1\) are exactly the central elements, because \[ |\operatorname{Cl}_G(x)|=1 \iff C_G(x)=G \iff x\in Z(G). \] Choosing one representative \(x_i\) from each noncentral conjugacy class gives the class equation \[ |G|=|Z(G)|+\sum_i [G:C_G(x_i)]. \]