I have a simple question on characters of simple groups; stated as exercise in Isaacs' character theory book.
(2.3) Let $\chi$ be a character of $G$. If $\mathfrak{X}$ is a representation corresponding to $\chi$, define $\det\chi(x):=\det\mathfrak{X}(x)$. Show: $\det\chi$ is well-defined linear character of $G$.
This is an easy exercise; consider the next one.
(3.3) Show that no simple group can have irreducible character of degree 2.
Hint: Problem 2.3 is relevant.
Question: My question is not about solution of (3.3) since I have solved it in some different way. I didn't click how (2.3) can be used for this?
Other way for (3.3): $|G|$ is divisible by $2$=degree of irreducible character. If $|G|=2.(odd)$ then it is well-known from basic group theory that $G$ can't be simple (Actually it is solvable). So $4$ should divide $|G|$. Consider subgroup $H$ of order $4$ in $G$. Let $\mathfrak{X}:G\rightarrow {\rm GL}_2(\mathbb{C})$ be irreducible with $G$ simple (non-abelian). We can show that some element $x\in H$, $x\neq 1$ goes to $\pm I$. So $Z(\chi)=\{g\in G: |\chi(g)|=\chi(1)\}$ is non-trivial normal subgroup. q.e.d.