Conjugacy class of prime power size implies not simple

Statement
If a finite group contains a fact about::conjugacy class whose size is a power of a prime (and also, greater than 1), then the finite group cannot be simple.

Applications
This is used to prove that order has only two prime factors implies solvable -- the result commonly called Burnside's $$p^aq^b$$-theorem.

Proof
Given: A finite group $$G$$, a prime number $$q$$, and a conjugacy class $$C$$ in $$G$$ of size $$q^d$$ where $$q$$ is a prime number and $$d > 0$$. $$g \in C$$ is a representative element.

To prove: $$G$$ is not a simple group.

Proof: Note that when the symbol $$1$$ appears as an input to a representation or a character, it refers to the identity element of $$G$$. When it appears as the output of a character, or in another context, it refers to the real number $$1$$.