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)].

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