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:
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,