No finite simple group has an irreducible character of degree two
= No finite simple group has an irreducible character of degree two
A degree-two irreducible representation of a nonabelian finite simple group would be faithful. Its determinant is a linear character and hence trivial, so its image lies in $\operatorname{SL}_2(\mathbb C)$. Degree divisibility makes the group order even; an involution must map to the unique nonidentity involution $-I$ in $\operatorname{SL}_2(\mathbb C)$ and would therefore be central, a contradiction.