Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 1 / c

The canonical image of $\Gamma$ is dense in its profinite completion $\hat{\Gamma }$. Therefore every $x\in \hat{\Gamma }$ is a limit of a net $({\delta }_{\lambda })$ in $\Gamma$. Continuity of conjugation givesso every element of ${\mathrm{Cl}}_{\hat{\Gamma }}(\gamma )$ lies in the closure of ${\mathrm{Cl}}_{\Gamma }(\gamma )$. The reverse inclusion follows because the larger conjugacy class contains the smaller one and is closed by part b(iii). Hence

Suppose first that $\Gamma$ is conjugacy separable. If $h\in \Gamma$ is not conjugate to $\gamma$ in $\Gamma$, some homomorphism to a finite group sends them to nonconjugate elements. This homomorphism factors through a finite quotient of $\hat{\Gamma }$, so $h$ cannot be conjugate to $\gamma$ in $\hat{\Gamma }$. Thus

Conversely, suppose this equality holds and $h$ is not conjugate to $\gamma$ in $\Gamma$. Then they are not conjugate in $\hat{\Gamma }$. By the finite-quotient criterion for conjugacy in a profinite group, their images fail to be conjugate in some finite quotient. This is precisely conjugacy separability.