= 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 $\rho(G)\subseteq\operatorname{SL}_2(\mathbb C)$.
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 $\operatorname{SL}_2(\mathbb C)$ 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 \b[no finite simple group has an irreducible character of degree two.]
Back to article page
