Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 5 / b / Solution

Let $\theta :H\wr G\to Q$ be any homomorphism to a finite group. Because $G$ is infinite, two distinct elements $r,s\in G$ have $\theta (r)=\theta (s)$. For $h\in H$, denote by $h_g$ the lamp with value $h$ at $g$. Conjugation translates lamps, sofor every $h\in H$. Choose $h,k\in H$ with $[h,k]\ne 1$. Lamps at different coordinates commute, and thereforeBut $[h_r,k_r]$ is the nonidentity lamp $[h,k{]}_r$. This same nonidentity element is killed by every finite quotient, so $H\wr G$ is not residually finite.