On a theorem of Burnside on endomorphisms

Question

I want to see some simple arguments for the following theorem of Burnside. I saw proof of this theorem in Jacobson's volume 2; but it was based on some enveloping algebra. Since in the theory of basic group representations, I had not came across enveloping algebra, I found the proof by Jacobson difficult to follow.

Theorem: Let $\rho:G\rightarrow {\rm GL}(V)$ be an irreducible representation of $G$ over $\mathbb{C}$. Then the set $\{\rho(g)\,|\, g\in G\}$ spans ${\rm End}(V)$.

Any simple proof of this?

Answer 1

A short and simple proof using nothing more than basic linear algebra is provided by T.Y. Lam here.