any $2$-dimensional rep of a finite, non-abelian simple group is trivial Let $G$ be a finite, non-abelian simple group. How would I go about proving that any $2$-dimensional representation of $G$ is trivial? If it helps, I know how to do it when we're considering $1$-dimensional representations.

This is exercise 3.3 in Character Theory of Finite Groups by Isaacs. It is easily deduced from the following facts:


*

*Because $G$ is perfect, $\chi$ has image in $SL(2,\mathbb{C})$;

*Because $\chi$ is irreducible, $\chi(1)$ divides $|G|$;

*The only element of order $2$ in $SL(2,\mathbb{C})$ is $-I$ (which is central).

