Class Equation of a Group

The class equation of a group is something that looks difficult at first sight, but is actually very straightforward once you understand it. An amazing equation…

Class Equation of a Group (Proof)

Suppose $G$ is a finite group, $Z(G)$ is the center of $G$ , and $c_1, c_2, \dots, c_r$ are all the conjugacy classes in $G$ comprising the elements outside the center. Let $g_i$ be an element in $c_i$ for each $1\leq i\leq r$ . Then we have: $\displaystyle |G|=|Z(G)|+\sum_{i=1}^r[G:C_G(g_i)].$

Proof:

Let $G$ act on itself by conjugation. The orbits of $G$ partition $G$ . Note that each conjugacy class $c_i$ is actually $\text{Orb}(g_i)$ .

Let $x\in Z(G)$ . Then $gxg^{-1}=xgg^{-1}=x$ for all $g\in G$ . Hence $\text{Orb}(x)$ consists of a single element $x$ itself.

Let $g_i\in c_i$ . Then
$\begin{aligned} \text{Stab}(g_i)&=\{h\in G\mid hg_ih^{-1}=g_i\}\\ &=\{h\in G\mid hg_i=g_ih\}\\ &=C_G(g_i). \end{aligned}$
By Orbit-Stabilizer Theorem, $\displaystyle |\text{Orb}(g_i)|=[G:\text{Stab}(g_i)]=[G:C_G(g_i)].$