Zero-or-scalar lemma

Over the complex numbers
Let $$G$$ be a finite group and $$\varphi$$ an fact about::irreducible linear representation of $$G$$ over $$\mathbb{C}$$. Let $$g \in G$$, such that the size of the conjugacy class of $$G$$ is relatively prime to the degree of $$\varphi$$. Then, either $$\varphi(g)$$ is a scalar or $$\chi(g) = 0$$.

Over a splitting field of characteristic zero
The proof as presented here works only over the complex numbers, but it can be generalized to any splitting field for $$G$$ that has characteristic zero.

Applications

 * Conjugacy class of prime power size implies not simple
 * Order has only two prime factors implies solvable, also called Burnside's $$p^aq^b$$-theorem (proved via conjugacy class of prime power size implies not simple)

Proof
Given: A finite group $$G$$, a nontrivial irreducible linear representation $$\varphi$$ of $$G$$ over $$\mathbb{C}$$ with character $$\chi$$. An element $$g \in G$$ with conjugacy class $$C$$. The degree of $$\varphi$$ and the size of $$C$$ are relatively prime.

To prove: Either $$\chi(g) = 0$$ or $$\varphi(g)$$ is a scalar.

Proof: To avoid confusion between the identity element of $$G$$ and the number 1, we will denote the identity element of $$G$$ by the letter $$e$$ rather than use the customary 1 to denote the identity element of $$G$$.