Zero-or-scalar lemma

Statement

Let $G$ be a finite group and $\phi$ a nontrivial Irreducible linear representation (?) of $G$. Let $g\in G$, such that the size of the conjugacy class of $G$ is relatively prime to the degree of $\phi$. Then, either $\phi (g)$ is a scalar or $\chi (g)=0$.

Facts used

1. Size-degree-weighted characters are algebraic integers

Proof

Let $C$ denote the conjugacy class of $g$. We use the fact that the following is an algebraic integer:

${\frac {|C|\chi (g)}{\chi (1)}}$

Now, if $\chi (1)$ and $|C|$ are relatively prime, then there exist integers $a$ and $b$ such that:

$a|C|+b\chi (1)=1$

multiplying both sides by $\chi (g)/\chi (1)$ we get:

${\frac {a|C|\chi (g)}{\chi (1)}}+b\chi (g)={\frac {\chi (g)}{\chi (1)}}$

The left-hand-side is an algebraic integer, hence so is the right-hand-side. Thus $\chi (g)/\chi (1)$ is an algebraic integer.

We know that $\chi (g)$ is a sum of $\chi (1)$ roots of unity (not necessarily all distinct). Thus, every algebraic conjugate of $\chi (g)$ is also a sum of $\chi (1)$ roots of unity. In particular, this tells us that the norm of $\chi (g)/\chi (1)$ is either $0$ or $\pm 1$.

Now if the norm is zero, then $\chi (g)=0$. If the norm is exactly 1, then all the roots of unity must be equal, and hence $\phi (g)$ must be a scalar matrix.