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 is a finite group, is the center of , and are all the conjugacy classes in comprising the elements outside the center. Let be an element in for each . Then we have:

## Proof:

Let act on itself by conjugation. The orbits of partition . Note that each conjugacy class is actually .

Let . Then for all . Hence consists of a single element itself.

Let . Then

By Orbit-Stabilizer Theorem,

Therefore,