Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 19I / iii / Solution

Suppose a finite simple group had an irreducible representation $\rho$ of degree two. Its kernel is normal and cannot be the whole group, so $\rho$ is faithful. A nonabelian simple group has no nontrivial linear character; consequently $\det \rho$ is trivial and $\rho (G)\subseteq {\mathrm{SL}}_2(\mathbb{C})$.

Since $2\mid |G|$ by degree divisibility, Cauchy's theorem for finite groups gives an involution $x\in G$. A nonidentity matrix of order two in ${\mathrm{SL}}_2(\mathbb{C})$ has both eigenvalues $-1$, hence equals $-I$. Thus $\rho (x)=-I$ is central in $\rho (G)$. Faithfulness makes $x$ central in $G$, contradicting the trivial centre of a nonabelian simple group. Therefore no finite simple group has an irreducible character of degree two.