Let prove that for any .

The conjugacy class equation says,(ii) By considering the Class Equation of G acting on itself by conjugation show that if ( p prime) then

.

Where the 's are taken from distinct conjugacy classes of more than one element.

We know that the left hand side is divisible by so the right hand side is divisible by . Now , the centralizer, is not whole because we are picking those , this means . This means divides the index of under . Thus, this proves that is divisible by . Thus, the center is non-trivial.