= Solution
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 $\widehat\Gamma$, so $h$ cannot be conjugate to $\gamma$ in $\widehat\Gamma$. Thus
$$
\operatorname{Cl}_{\widehat\Gamma}(\gamma)\cap\Gamma
=\operatorname{Cl}_\Gamma(\gamma).
$$
Conversely, suppose this equality holds and $h$ is not conjugate to $\gamma$ in $\Gamma$. Then they are not conjugate in $\widehat\Gamma$. 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.
Back to article page
