Prove that the size of a conjugacy class of \(g_i\) is the index of its centralizer, \([G: Z(g_i)] \da [G: C_G(g_i)]\).
Solution
Let \(G\) act on itself by conjugation: \[ h\cdot x=hxh^{-1}. \] The orbit of \(x\) is its conjugacy class \[ \operatorname{Cl}_G(x)=\{hxh^{-1}:h\in G\}, \] and its stabilizer is \[ \operatorname{Stab}_G(x) =\{h:hxh^{-1}=x\} =C_G(x). \]
The orbit-stabilizer bijection is explicitly \[ G/C_G(x)\longrightarrow \operatorname{Cl}_G(x), \qquad hC_G(x)\longmapsto hxh^{-1}. \] Indeed, \[ hxh^{-1}=kxk^{-1} \iff k^{-1}h\in C_G(x) \iff hC_G(x)=kC_G(x). \] Thus \[ \boxed{|\operatorname{Cl}_G(x)|=[G:C_G(x)]}. \] For finite \(G\), this is \(|G|/|C_G(x)|\).