The question is essentially the title. We assume that $G$ is a finite group and the irrep is over $\mathbb C$. $Z(G)$ is the center of $G$.
The caveat is that I'm looking for an "elementary" proof. I've found this already, which is proving a more general theorem, which uses induced representations and isotypical representations.
My question whether it's possible to prove the result without using these tools, and using only "elementary" results from representation theory.