This should be a straightforward calculation.Prove is a subgroup.
Again a straightforward calculation, just look at definition of group action.Show defines an action on itself.
This is just the conjugacy class equation.
Here , i.e. the group center. While . And this follows because is a sum which goes through all the non-trivial conjugacy classes, i.e. when the orbit is greater than 1.
This result is known as Burnside's Lemma.Show
Now since it means . Thus, the sum is divisible by and is divisible by . It must mean that is divisible by .