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 ${\mathrm{SL}}_2(\mathbb{C})$. Degree divisibility makes the group order even; an involution must map to the unique nonidentity involution $-I$ in ${\mathrm{SL}}_2(\mathbb{C})$ and would therefore be central, a contradiction.