Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 19F / a / Solution

The kernel is $\{g:\rho (g)=I\}$ and is normal because $\rho (hg{h}^{-1})=\rho (h)\rho (g)\rho (h{)}^{-1}$. A one-dimensional representation kills every commutator. Conversely, irreducibles killing $G^{\prime }$ factor through the finite abelian group $G/G^{\prime }$ and are one-dimensional.