# 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, $\displaystyle |\text{Orb}(g_i)|=[G:\text{Stab}(g_i)]=[G:C_G(g_i)].$

Therefore, $\displaystyle |G|=|Z(G)|+\sum_{i=1}^r[G:C_G(g_i)].$

## Author: mathtuition88

https://mathtuition88.com/

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